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# Cyclotomic polynomial

Cyclotomic polynomial
You are encouraged to solve this task according to the task description, using any language you may know.

The nth Cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial of largest degree with integer coefficients that is a divisor of x^n − 1, and is not a divisor of x^k − 1 for any k < n.

• Find and print the first 30 cyclotomic polynomials.
• Find and print the order of the first 10 cyclotomic polynomials that have n or -n as a coefficient.

• Wikipedia article, Cyclotomic polynomial, showing ways to calculate them.
• The sequence A013594 with the smallest order of cyclotomic polynomial containing n or -n as a coefficient.

## C++

Translation of: Java
`#include <algorithm>#include <iostream>#include <initializer_list>#include <map>#include <vector> const int MAX_ALL_FACTORS = 100000;const int algorithm = 2;int divisions = 0; //Note: Cyclotomic Polynomials have small coefficients.  Not appropriate for general polynomial usage.class Term {private:    long m_coefficient;    long m_exponent; public:    Term(long c, long e) : m_coefficient(c), m_exponent(e) {        // empty    }     Term(const Term &t) : m_coefficient(t.m_coefficient), m_exponent(t.m_exponent) {        // empty    }     long coefficient() const {        return m_coefficient;    }     long degree() const {        return m_exponent;    }     Term operator -() const {        return { -m_coefficient, m_exponent };    }     Term operator *(const Term &rhs) const {        return { m_coefficient * rhs.m_coefficient, m_exponent + rhs.m_exponent };    }     Term operator +(const Term &rhs) const {        if (m_exponent != rhs.m_exponent) {            throw std::runtime_error("Exponents not equal");        }        return { m_coefficient + rhs.m_coefficient, m_exponent };    }     friend std::ostream &operator<<(std::ostream &, const Term &);}; std::ostream &operator<<(std::ostream &os, const Term &t) {    if (t.m_coefficient == 0) {        return os << '0';    }    if (t.m_exponent == 0) {        return os << t.m_coefficient;    }    if (t.m_coefficient == 1) {        if (t.m_exponent == 1) {            return os << 'x';        }        return os << "x^" << t.m_exponent;    }    if (t.m_coefficient == -1) {        if (t.m_exponent == 1) {            return os << "-x";        }        return os << "-x^" << t.m_exponent;    }    if (t.m_exponent == 1) {        return os << t.m_coefficient << 'x';    }    return os << t.m_coefficient << "x^" << t.m_exponent;} class Polynomial {public:    std::vector<Term> polynomialTerms;     Polynomial() {        polynomialTerms.push_back({ 0, 0 });    }     Polynomial(std::initializer_list<int> values) {        if (values.size() % 2 != 0) {            throw std::runtime_error("Length must be even.");        }         bool ready = false;        long t;        for (auto v : values) {            if (ready) {                polynomialTerms.push_back({ t, v });            } else {                t = v;            }            ready = !ready;        }        std::sort(            polynomialTerms.begin(), polynomialTerms.end(),            [](const Term &t, const Term &u) {                return u.degree() < t.degree();            }        );    }     Polynomial(const std::vector<Term> &termList) {        if (termList.size() == 0) {            polynomialTerms.push_back({ 0, 0 });        } else {            for (auto t : termList) {                if (t.coefficient() != 0) {                    polynomialTerms.push_back(t);                }            }            if (polynomialTerms.size() == 0) {                polynomialTerms.push_back({ 0, 0 });            }            std::sort(                polynomialTerms.begin(), polynomialTerms.end(),                [](const Term &t, const Term &u) {                    return u.degree() < t.degree();                }            );        }    }     Polynomial(const Polynomial &p) : Polynomial(p.polynomialTerms) {        // empty    }     long leadingCoefficient() const {        return polynomialTerms[0].coefficient();    }     long degree() const {        return polynomialTerms[0].degree();    }     bool hasCoefficientAbs(int coeff) {        for (auto term : polynomialTerms) {            if (abs(term.coefficient()) == coeff) {                return true;            }        }        return false;    }     Polynomial operator+(const Term &term) const {        std::vector<Term> termList;        bool added = false;        for (size_t index = 0; index < polynomialTerms.size(); index++) {            auto currentTerm = polynomialTerms[index];            if (currentTerm.degree() == term.degree()) {                added = true;                if (currentTerm.coefficient() + term.coefficient() != 0) {                    termList.push_back(currentTerm + term);                }            } else {                termList.push_back(currentTerm);            }        }        if (!added) {            termList.push_back(term);        }        return Polynomial(termList);    }     Polynomial operator*(const Term &term) const {        std::vector<Term> termList;        for (size_t index = 0; index < polynomialTerms.size(); index++) {            auto currentTerm = polynomialTerms[index];            termList.push_back(currentTerm * term);        }        return Polynomial(termList);    }     Polynomial operator+(const Polynomial &p) const {        std::vector<Term> termList;        int thisCount = polynomialTerms.size();        int polyCount = p.polynomialTerms.size();        while (thisCount > 0 || polyCount > 0) {            if (thisCount == 0) {                auto polyTerm = p.polynomialTerms[polyCount - 1];                termList.push_back(polyTerm);                polyCount--;            } else if (polyCount == 0) {                auto thisTerm = polynomialTerms[thisCount - 1];                termList.push_back(thisTerm);                thisCount--;            } else {                auto polyTerm = p.polynomialTerms[polyCount - 1];                auto thisTerm = polynomialTerms[thisCount - 1];                if (thisTerm.degree() == polyTerm.degree()) {                    auto t = thisTerm + polyTerm;                    if (t.coefficient() != 0) {                        termList.push_back(t);                    }                    thisCount--;                    polyCount--;                } else if (thisTerm.degree() < polyTerm.degree()) {                    termList.push_back(thisTerm);                    thisCount--;                } else {                    termList.push_back(polyTerm);                    polyCount--;                }            }        }        return Polynomial(termList);    }     Polynomial operator/(const Polynomial &v) {        divisions++;         Polynomial q;        Polynomial r(*this);        long lcv = v.leadingCoefficient();        long dv = v.degree();        while (r.degree() >= v.degree()) {            long lcr = r.leadingCoefficient();            long s = lcr / lcv;            Term term(s, r.degree() - dv);            q = q + term;            r = r + v * -term;        }         return q;    }     friend std::ostream &operator<<(std::ostream &, const Polynomial &);}; std::ostream &operator<<(std::ostream &os, const Polynomial &p) {    auto it = p.polynomialTerms.cbegin();    auto end = p.polynomialTerms.cend();    if (it != end) {        os << *it;        it = std::next(it);    }    while (it != end) {        if (it->coefficient() > 0) {            os << " + " << *it;        } else {            os << " - " << -*it;        }        it = std::next(it);    }    return os;} std::vector<int> getDivisors(int number) {    std::vector<int> divisiors;    long root = (long)sqrt(number);    for (int i = 1; i <= root; i++) {        if (number % i == 0) {            divisiors.push_back(i);            int div = number / i;            if (div != i && div != number) {                divisiors.push_back(div);            }        }    }    return divisiors;} std::map<int, std::map<int, int>> allFactors; std::map<int, int> getFactors(int number) {    if (allFactors.find(number) != allFactors.end()) {        return allFactors[number];    }     std::map<int, int> factors;    if (number % 2 == 0) {        auto factorsDivTwo = getFactors(number / 2);        factors.insert(factorsDivTwo.begin(), factorsDivTwo.end());        if (factors.find(2) != factors.end()) {            factors[2]++;        } else {            factors.insert(std::make_pair(2, 1));        }        if (number < MAX_ALL_FACTORS) {            allFactors.insert(std::make_pair(number, factors));        }        return factors;    }    long root = (long)sqrt(number);    long i = 3;    while (i <= root) {        if (number % i == 0) {            auto factorsDivI = getFactors(number / i);            factors.insert(factorsDivI.begin(), factorsDivI.end());            if (factors.find(i) != factors.end()) {                factors[i]++;            } else {                factors.insert(std::make_pair(i, 1));            }            if (number < MAX_ALL_FACTORS) {                allFactors.insert(std::make_pair(number, factors));            }            return factors;        }        i += 2;    }    factors.insert(std::make_pair(number, 1));    if (number < MAX_ALL_FACTORS) {        allFactors.insert(std::make_pair(number, factors));    }    return factors;} std::map<int, Polynomial> COMPUTED;Polynomial cyclotomicPolynomial(int n) {    if (COMPUTED.find(n) != COMPUTED.end()) {        return COMPUTED[n];    }     if (n == 1) {        // Polynomial: x - 1        Polynomial p({ 1, 1, -1, 0 });        COMPUTED.insert(std::make_pair(1, p));        return p;    }     auto factors = getFactors(n);    if (factors.find(n) != factors.end()) {        // n prime        std::vector<Term> termList;        for (int index = 0; index < n; index++) {            termList.push_back({ 1, index });        }         Polynomial cyclo(termList);        COMPUTED.insert(std::make_pair(n, cyclo));        return cyclo;    } else if (factors.size() == 2 && factors.find(2) != factors.end() && factors[2] == 1 && factors.find(n / 2) != factors.end() && factors[n / 2] == 1) {        // n = 2p        int prime = n / 2;        std::vector<Term> termList;        int coeff = -1;         for (int index = 0; index < prime; index++) {            coeff *= -1;            termList.push_back({ coeff, index });        }         Polynomial cyclo(termList);        COMPUTED.insert(std::make_pair(n, cyclo));        return cyclo;    } else if (factors.size() == 1 && factors.find(2) != factors.end()) {        // n = 2^h        int h = factors[2];        std::vector<Term> termList;        termList.push_back({ 1, (int)pow(2, h - 1) });        termList.push_back({ 1, 0 });         Polynomial cyclo(termList);        COMPUTED.insert(std::make_pair(n, cyclo));        return cyclo;    } else if (factors.size() == 1 && factors.find(n) != factors.end()) {        // n = p^k        int p = 0;        int k = 0;        for (auto iter = factors.begin(); iter != factors.end(); ++iter) {            p = iter->first;            k = iter->second;        }        std::vector<Term> termList;        for (int index = 0; index < p; index++) {            termList.push_back({ 1, index * (int)pow(p, k - 1) });        }         Polynomial cyclo(termList);        COMPUTED.insert(std::make_pair(n, cyclo));        return cyclo;    } else if (factors.size() == 2 && factors.find(2) != factors.end()) {        // n = 2^h * p^k        int p = 0;        for (auto iter = factors.begin(); iter != factors.end(); ++iter) {            if (iter->first != 2) {                p = iter->first;            }        }         std::vector<Term> termList;        int coeff = -1;        int twoExp = (int)pow(2, factors[2] - 1);        int k = factors[p];        for (int index = 0; index < p; index++) {            coeff *= -1;            termList.push_back({ coeff, index * twoExp * (int)pow(p, k - 1) });        }         Polynomial cyclo(termList);        COMPUTED.insert(std::make_pair(n, cyclo));        return cyclo;    } else if (factors.find(2) != factors.end() && ((n / 2) % 2 == 1) && (n / 2) > 1) {        //  CP(2m)[x] = CP(-m)[x], n odd integer > 1        auto cycloDiv2 = cyclotomicPolynomial(n / 2);        std::vector<Term> termList;        for (auto term : cycloDiv2.polynomialTerms) {            if (term.degree() % 2 == 0) {                termList.push_back(term);            } else {                termList.push_back(-term);            }        }         Polynomial cyclo(termList);        COMPUTED.insert(std::make_pair(n, cyclo));        return cyclo;    }     // General Case     if (algorithm == 0) {        // slow - uses basic definition        auto divisors = getDivisors(n);        // Polynomial: (x^n - 1)        Polynomial cyclo({ 1, n, -1, 0 });        for (auto i : divisors) {            auto p = cyclotomicPolynomial(i);            cyclo = cyclo / p;        }         COMPUTED.insert(std::make_pair(n, cyclo));        return cyclo;    } else if (algorithm == 1) {        //  Faster.  Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor        auto divisors = getDivisors(n);        int maxDivisor = INT_MIN;        for (auto div : divisors) {            maxDivisor = std::max(maxDivisor, div);        }        std::vector<int> divisorExceptMax;        for (auto div : divisors) {            if (maxDivisor % div != 0) {                divisorExceptMax.push_back(div);            }        }         //  Polynomial:  ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor        auto cyclo = Polynomial({ 1, n, -1, 0 }) / Polynomial({ 1, maxDivisor, -1, 0 });        for (int i : divisorExceptMax) {            auto p = cyclotomicPolynomial(i);            cyclo = cyclo / p;        }         COMPUTED.insert(std::make_pair(n, cyclo));        return cyclo;    } else if (algorithm == 2) {        //  Fastest        //  Let p ; q be primes such that p does not divide n, and q q divides n.        //  Then CP(np)[x] = CP(n)[x^p] / CP(n)[x]        int m = 1;        auto cyclo = cyclotomicPolynomial(m);        std::vector<int> primes;        for (auto iter = factors.begin(); iter != factors.end(); ++iter) {            primes.push_back(iter->first);        }        std::sort(primes.begin(), primes.end());        for (auto prime : primes) {            //  CP(m)[x]            auto cycloM = cyclo;            //  Compute CP(m)[x^p].            std::vector<Term> termList;            for (auto t : cycloM.polynomialTerms) {                termList.push_back({ t.coefficient(), t.degree() * prime });            }            cyclo = Polynomial(termList) / cycloM;            m = m * prime;        }        //  Now, m is the largest square free divisor of n        int s = n / m;        //  Compute CP(n)[x] = CP(m)[x^s]        std::vector<Term> termList;        for (auto t : cyclo.polynomialTerms) {            termList.push_back({ t.coefficient(), t.degree() * s });        }         cyclo = Polynomial(termList);        COMPUTED.insert(std::make_pair(n, cyclo));        return cyclo;    } else {        throw std::runtime_error("Invalid algorithm");    }} int main() {    // initialization    std::map<int, int> factors;    factors.insert(std::make_pair(2, 1));    allFactors.insert(std::make_pair(2, factors));     // rest of main    std::cout << "Task 1:  cyclotomic polynomials for n <= 30:\n";    for (int i = 1; i <= 30; i++) {        auto p = cyclotomicPolynomial(i);        std::cout << "CP[" << i << "] = " << p << '\n';    }     std::cout << "Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:\n";    int n = 0;    for (int i = 1; i <= 10; i++) {        while (true) {            n++;            auto cyclo = cyclotomicPolynomial(n);            if (cyclo.hasCoefficientAbs(i)) {                std::cout << "CP[" << n << "] has coefficient with magnitude = " << i << '\n';                n--;                break;            }        }    }     return 0;}`
Output:
```Task 1:  cyclotomic polynomials for n <= 30:
CP[1] = x - 1
CP[2] = x + 1
CP[3] = x^2 + x + 1
CP[4] = x^2 + 1
CP[5] = x^4 + x^3 + x^2 + x + 1
CP[6] = x^2 - x + 1
CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[8] = x^4 + 1
CP[9] = x^6 + x^3 + 1
CP[10] = x^4 - x^3 + x^2 - x + 1
CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[12] = x^4 - x^2 + 1
CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
CP[16] = x^8 + 1
CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[18] = x^6 - x^3 + 1
CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[20] = x^8 - x^6 + x^4 - x^2 + 1
CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[24] = x^8 - x^4 + 1
CP[25] = x^20 + x^15 + x^10 + x^5 + 1
CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[27] = x^18 + x^9 + 1
CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:
CP[1] has coefficient with magnitude = 1
CP[105] has coefficient with magnitude = 2
CP[385] has coefficient with magnitude = 3
CP[1365] has coefficient with magnitude = 4
CP[1785] has coefficient with magnitude = 5
CP[2805] has coefficient with magnitude = 6
CP[3135] has coefficient with magnitude = 7
CP[6545] has coefficient with magnitude = 8
CP[6545] has coefficient with magnitude = 9
CP[10465] has coefficient with magnitude = 10```

## C#

Translation of: Java
Works with: C sharp version 8
`using System;using System.Collections;using System.Collections.Generic;using System.Linq;using IntMap = System.Collections.Generic.Dictionary<int, int>; public static class CyclotomicPolynomial{    public static void Main2() {        Console.WriteLine("Task 1: Cyclotomic polynomials for n <= 30:");        for (int i = 1; i <= 30; i++) {            var p = GetCyclotomicPolynomial(i);            Console.WriteLine(\$"CP[{i}] = {p.ToString()}");        }        Console.WriteLine();         Console.WriteLine("Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:");        for (int i = 1, n = 0; i <= 10; i++) {            while (true) {                n++;                var p = GetCyclotomicPolynomial(n);                if (p.Any(t => Math.Abs(t.Coefficient) == i)) {                    Console.WriteLine(\$"CP[{n}] has coefficient with magnitude = {i}");                    n--;                    break;                }            }        }    }     private const int MaxFactors = 100_000;    private const int Algorithm = 2;    private static readonly Term x = new Term(1, 1);    private static readonly Dictionary<int, Polynomial> polyCache =        new Dictionary<int, Polynomial> { [1] = x - 1 };    private static readonly Dictionary<int, IntMap> factorCache =        new Dictionary<int, IntMap> { [2] = new IntMap { [2] = 1 } };     private static Polynomial GetCyclotomicPolynomial(in int n) {        if (polyCache.TryGetValue(n, out var result)) return result;         var factors = GetFactors(n);        if (factors.ContainsKey(n)) { //n is prime            result = new Polynomial(from exp in ..n select x[exp]);        } else if (factors.Count == 2 && factors.Contains(2, 1) && factors.Contains(n/2, 1)) { //n = 2p            result = new Polynomial(from i in ..(n/2) select (IsOdd(i) ? -x : x)[i]);        } else if (factors.Count == 1 && factors.TryGetValue(2, out int h)) { //n = 2^h            result = x[1<<(h-1)] + 1;        } else if (factors.Count == 1 && !factors.ContainsKey(n)) { // n = p^k            (int p, int k) = factors.First();            result = new Polynomial(from i in ..p select x[i * (int)Math.Pow(p, k-1)]);        } else if (factors.Count == 2 && factors.ContainsKey(2)) { //n = 2^h * p^k            (int p, int k) = factors.First(entry => entry.Key != 2);            int twoExp = 1 << (factors[2] - 1);            result = new Polynomial(from i in ..p select (IsOdd(i) ? -x : x)[i * twoExp * (int)Math.Pow(p, k-1)]);        } else if (factors.ContainsKey(2) && IsOdd(n/2) && n/2 > 1) { // CP(2m)[x] = CP[-m][x], n is odd > 1            Polynomial cycloDiv2 = GetCyclotomicPolynomial(n/2);            result = new Polynomial(from term in cycloDiv2 select IsOdd(term.Exponent) ? -term : term);            #pragma warning disable CS0162        } else if (Algorithm == 0) {            var divisors = GetDivisors(n);            result = x[n] - 1;            foreach (int d in divisors) result /= GetCyclotomicPolynomial(d);        } else if (Algorithm == 1) {            var divisors = GetDivisors(n).ToList();            int maxDivisor = divisors.Max();            result = (x[n] - 1) / (x[maxDivisor] - 1);            foreach (int d in divisors.Where(div => maxDivisor % div == 0)) {                result /= GetCyclotomicPolynomial(d);            }        } else if (Algorithm == 2) {            int m = 1;            result = GetCyclotomicPolynomial(m);            var primes = factors.Keys.ToList();            primes.Sort();            foreach (int prime in primes) {                var cycloM = result;                result = new Polynomial(from term in cycloM select term.Coefficient * x[term.Exponent * prime]);                result /= cycloM;                m *= prime;            }            int s = n / m;            result = new Polynomial(from term in result select term.Coefficient * x[term.Exponent * s]);            #pragma warning restore CS0162        } else {            throw new InvalidOperationException("Invalid algorithm");        }        polyCache[n] = result;        return result;    }     private static bool IsOdd(int i) => (i & 1) != 0;    private static bool Contains(this IntMap map, int key, int value) => map.TryGetValue(key, out int v) && v == value;    private static int GetOrZero(this IntMap map, int key) => map.TryGetValue(key, out int v) ? v : 0;    private static IEnumerable<T> Select<T>(this Range r, Func<int, T> f) => Enumerable.Range(r.Start.Value, r.End.Value - r.Start.Value).Select(f);     private static IntMap GetFactors(in int n) {        if (factorCache.TryGetValue(n, out var factors)) return factors;         factors = new IntMap();        if (!IsOdd(n)) {            foreach (var entry in GetFactors(n/2)) factors.Add(entry.Key, entry.Value);            factors[2] = factors.GetOrZero(2) + 1;            return Cache(n, factors);        }        for (int i = 3; i * i <= n; i+=2) {            if (n % i == 0) {                foreach (var entry in GetFactors(n/i)) factors.Add(entry.Key, entry.Value);                factors[i] = factors.GetOrZero(i) + 1;                return Cache(n, factors);            }        }        factors[n] = 1;        return Cache(n, factors);    }     private static IntMap Cache(int n, IntMap factors) {        if (n < MaxFactors) factorCache[n] = factors;        return factors;    }     private static IEnumerable<int> GetDivisors(int n) {        for (int i = 1; i * i <= n; i++) {            if (n % i == 0) {                yield return i;                int div = n / i;                if (div != i && div != n) yield return div;            }        }    }     public sealed class Polynomial : IEnumerable<Term>    {        public Polynomial() { }        public Polynomial(params Term[] terms) : this(terms.AsEnumerable()) { }         public Polynomial(IEnumerable<Term> terms) {            Terms.AddRange(terms);            Simplify();        }         private List<Term>? terms;        private List<Term> Terms => terms ??= new List<Term>();         public int Count => terms?.Count ?? 0;        public int Degree => Count == 0 ? -1 : Terms[0].Exponent;        public int LeadingCoefficient => Count == 0 ? 0 : Terms[0].Coefficient;         public IEnumerator<Term> GetEnumerator() => Terms.GetEnumerator();        IEnumerator IEnumerable.GetEnumerator() => GetEnumerator();         public override string ToString() => Count == 0 ? "0" : string.Join(" + ", Terms).Replace("+ -", "- ");         public static Polynomial operator *(Polynomial p, Term t) => new Polynomial(from s in p select s * t);        public static Polynomial operator +(Polynomial p, Polynomial q) => new Polynomial(p.Terms.Concat(q.Terms));        public static Polynomial operator -(Polynomial p, Polynomial q) => new Polynomial(p.Terms.Concat(q.Terms.Select(t => -t)));        public static Polynomial operator *(Polynomial p, Polynomial q) => new Polynomial(from s in p from t in q select s * t);        public static Polynomial operator /(Polynomial p, Polynomial q) => p.Divide(q).quotient;         public (Polynomial quotient, Polynomial remainder) Divide(Polynomial divisor) {            if (Degree < 0) return (new Polynomial(), this);            Polynomial quotient = new Polynomial();            Polynomial remainder = this;            int lcv = divisor.LeadingCoefficient;            int dv = divisor.Degree;            while (remainder.Degree >= divisor.Degree) {                int lcr = remainder.LeadingCoefficient;                Term div = new Term(lcr / lcv, remainder.Degree - dv);                quotient.Terms.Add(div);                remainder += divisor * -div;            }            quotient.Simplify();            remainder.Simplify();            return (quotient, remainder);        }         private void Simplify() {            if (Count < 2) return;            Terms.Sort((a, b) => -a.CompareTo(b));            for (int i = Terms.Count - 1; i > 0; i--) {                Term s = Terms[i-1];                Term t = Terms[i];                if (t.Exponent == s.Exponent) {                    Terms[i-1] = new Term(s.Coefficient + t.Coefficient, s.Exponent);                    Terms.RemoveAt(i);                }            }            Terms.RemoveAll(t => t.IsZero);        }     }     public readonly struct Term : IEquatable<Term>, IComparable<Term>    {        public Term(int coefficient, int exponent = 0) => (Coefficient, Exponent) = (coefficient, exponent);         public Term this[int exponent] => new Term(Coefficient, exponent); //Using x[exp] because x^exp has low precedence        public int Coefficient { get; }        public int Exponent { get; }        public bool IsZero => Coefficient == 0;         public static Polynomial operator +(Term left, Term right) => new Polynomial(left, right);        public static Polynomial operator -(Term left, Term right) => new Polynomial(left, -right);        public static implicit operator Term(int coefficient) => new Term(coefficient);        public static Term operator -(Term t) => new Term(-t.Coefficient, t.Exponent);        public static Term operator *(Term left, Term right) => new Term(left.Coefficient * right.Coefficient, left.Exponent + right.Exponent);         public static bool operator ==(Term left, Term right) => left.Equals(right);        public static bool operator !=(Term left, Term right) => !left.Equals(right);        public static bool operator  <(Term left, Term right) => left.CompareTo(right)  < 0;        public static bool operator  >(Term left, Term right) => left.CompareTo(right)  > 0;        public static bool operator <=(Term left, Term right) => left.CompareTo(right) <= 0;        public static bool operator >=(Term left, Term right) => left.CompareTo(right) >= 0;         public bool Equals(Term other) => Exponent == other.Exponent && Coefficient == other.Coefficient;        public override bool Equals(object? obj) => obj is Term t && Equals(t);        public override int GetHashCode() => Coefficient.GetHashCode() * 31 + Exponent.GetHashCode();         public int CompareTo(Term other) {            int c = Exponent.CompareTo(other.Exponent);            if (c != 0) return c;            return Coefficient.CompareTo(other.Coefficient);        }         public override string ToString() => (Coefficient, Exponent) switch {            (0,  _) => "0",            (_,  0) => \$"{Coefficient}",            (1,  1) => "x",            (-1, 1) => "-x",            (_,  1) => \$"{Coefficient}x",            (1,  _) => \$"x^{Exponent}",            (-1, _) => \$"-x^{Exponent}",                    _ => \$"{Coefficient}x^{Exponent}"        };    }}`
Output:
```Task 1: Cyclotomic polynomials for n <= 30:
CP[1] = x - 1
CP[2] = x + 1
CP[3] = x^2 + x + 1
CP[4] = x^2 + 1
CP[5] = x^4 + x^3 + x^2 + x + 1
CP[6] = x^2 - x + 1
CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[8] = x^4 + 1
CP[9] = x^6 + x^3 + 1
CP[10] = x^4 - x^3 + x^2 - x + 1
CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[12] = x^4 - x^2 + 1
CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
CP[16] = x^8 + 1
CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[18] = x^6 - x^3 + 1
CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[20] = x^8 - x^6 + x^4 - x^2 + 1
CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[24] = x^8 - x^4 + 1
CP[25] = x^20 + x^15 + x^10 + x^5 + 1
CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[27] = x^18 + x^9 + 1
CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Task 2: Smallest cyclotomic polynomial with n or -n as a coefficient:
CP[1] has coefficient with magnitude = 1
CP[105] has coefficient with magnitude = 2
CP[385] has coefficient with magnitude = 3
CP[1365] has coefficient with magnitude = 4
CP[1785] has coefficient with magnitude = 5
CP[2805] has coefficient with magnitude = 6
CP[3135] has coefficient with magnitude = 7
CP[6545] has coefficient with magnitude = 8
CP[6545] has coefficient with magnitude = 9
CP[10465] has coefficient with magnitude = 10```

## D

Translation of: Kotlin
`import std.algorithm;import std.exception;import std.format;import std.functional;import std.math;import std.range;import std.stdio; immutable MAX_ALL_FACTORS = 100_000;immutable ALGORITHM = 2; //Note: Cyclotomic Polynomials have small coefficients.  Not appropriate for general polynomial usage. struct Term {    private long m_coefficient;    private long m_exponent;     public this(long c, long e) {        m_coefficient = c;        m_exponent = e;    }     public long coefficient() const {        return m_coefficient;    }     public long exponent() const {        return m_exponent;    }     public Term opUnary(string op)() const {        static if (op == "-") {            return Term(-m_coefficient, m_exponent);        } else {            assert(false, "Not implemented");        }    }     public Term opBinary(string op)(Term term) const {        static if (op == "+") {            if (exponent() != term.exponent()) {                assert(false, "Error 102: Exponents not equals.");            }            return Term(coefficient() + term.coefficient(), exponent());        } else if (op == "*") {            return Term(coefficient() * term.coefficient(), exponent() + term.exponent());        } else {            assert(false, "Not implemented: " ~ op);        }    }     public void toString(scope void delegate(const(char)[]) sink) const {        auto spec = singleSpec("%s");        if (m_coefficient == 0) {            sink("0");        } else if (m_exponent == 0) {            formatValue(sink, m_coefficient, spec);        } else if (m_coefficient == 1) {            if (m_exponent == 1) {                sink("x");            } else {                sink("x^");                formatValue(sink, m_exponent, spec);            }        } else if (m_coefficient == -1) {            if (m_exponent == 1) {                sink("-x");            } else {                sink("-x^");                formatValue(sink, m_exponent, spec);            }        } else if (m_exponent == 1) {            formatValue(sink, m_coefficient, spec);            sink("x");        } else {            formatValue(sink, m_coefficient, spec);            sink("x^");            formatValue(sink, m_exponent, spec);        }    }} struct Polynomial {    private Term[] terms;     public this(const Term[] ts...) {        terms = ts.dup;        terms.sort!"b.exponent < a.exponent";    }     bool hasCoefficientAbs(int coeff) const {        foreach (term; terms) {            if (abs(term.coefficient) == coeff) {                return true;            }        }        return false;    }     public long leadingCoefficient() const {        return terms[0].coefficient();    }     public long degree() const {        if (terms.empty) {            return -1;        }        return terms[0].exponent();    }     public Polynomial opBinary(string op)(Term term) const {        static if (op == "+") {            Term[] newTerms;            auto added = false;            foreach (currentTerm; terms) {                if (currentTerm.exponent() == term.exponent()) {                    added = true;                    if (currentTerm.coefficient() + term.coefficient() != 0) {                        newTerms ~= currentTerm + term;                    }                } else {                    newTerms ~= currentTerm;                }            }            if (!added) {                newTerms ~= term;            }            return Polynomial(newTerms);        } else if (op == "*") {            Term[] newTerms;            foreach (currentTerm; terms) {                newTerms ~= currentTerm * term;            }            return Polynomial(newTerms);        } else {            assert(false, "Not implemented: " ~ op);        }    }     public Polynomial opBinary(string op)(Polynomial rhs) const {        static if (op == "+") {            Term[] newTerms;            auto thisCount = terms.length;            auto polyCount = rhs.terms.length;            while (thisCount > 0 || polyCount > 0) {                if (thisCount == 0) {                    newTerms ~= rhs.terms[polyCount - 1];                    polyCount--;                } else if (polyCount == 0) {                    newTerms ~= terms[thisCount - 1];                    thisCount--;                } else {                    auto thisTerm = terms[thisCount - 1];                    auto polyTerm = rhs.terms[polyCount - 1];                    if (thisTerm.exponent() == polyTerm.exponent()) {                        auto t = thisTerm + polyTerm;                        if (t.coefficient() != 0) {                            newTerms ~= t;                        }                        thisCount--;                        polyCount--;                    } else if (thisTerm.exponent() < polyTerm.exponent()) {                        newTerms ~= thisTerm;                        thisCount--;                    } else {                        newTerms ~= polyTerm;                        polyCount--;                    }                }            }            return Polynomial(newTerms);        } else if (op == "/") {            Polynomial q;            auto r = Polynomial(terms);            auto lcv = rhs.leadingCoefficient();            auto dv = rhs.degree();            while (r.degree() >= rhs.degree()) {                auto lcr = r.leadingCoefficient();                auto s = lcr / lcv;                auto term = Term(s, r.degree() - dv);                q = q + term;                r = r + rhs * -term;            }            return q;        } else {            assert(false, "Not implemented: " ~ op);        }    }     public int opApply(int delegate(Term) dg) const {        foreach (term; terms) {            auto rv = dg(term);            if (rv != 0) {                return rv;            }        }        return 0;    }     public void toString(scope void delegate(const(char)[]) sink) const {        auto spec = singleSpec("%s");        if (!terms.empty) {            formatValue(sink, terms[0], spec);            foreach (t; terms[1..\$]) {                if (t.coefficient > 0) {                    sink(" + ");                    formatValue(sink, t, spec);                } else {                    sink(" - ");                    formatValue(sink, -t, spec);                }            }        }    }} void putAll(K, V)(ref V[K] a, V[K] b) {    foreach (k, v; b) {        a[k] = v;    }} void merge(K, V, F)(ref V[K] a, K k, V v, F f) {    if (k in a) {        a[k] = f(a[k], v);    } else {        a[k] = v;    }} int sum(int a, int b) {    return a + b;} int[int] getFactorsImpl(int number) {    int[int] factors;    if (number % 2 == 0) {        if (number > 2) {            auto factorsDivTwo = memoize!getFactorsImpl(number / 2);            factors.putAll(factorsDivTwo);        }        factors.merge(2, 1, &sum);        return factors;    }    auto root = sqrt(cast(real) number);    auto i = 3;    while (i <= root) {        if (number % i == 0) {            factors.putAll(memoize!getFactorsImpl(number / i));            factors.merge(i, 1, &sum);            return factors;        }        i += 2;    }    factors[number] = 1;    return factors;}alias getFactors = memoize!getFactorsImpl; int[] getDivisors(int number) {    int[] divisors;    auto root = cast(int)sqrt(cast(real) number);    foreach (i; 1..root) {        if (number % i == 0) {            divisors ~= i;        }        auto div = number / i;        if (div != i && div != number) {            divisors ~= div;        }    }    return divisors;} Polynomial cyclotomicPolynomialImpl(int n) {    if (n == 1) {        //  Polynomial:  x - 1        return Polynomial(Term(1, 1), Term(-1, 0));    }    auto factors = getFactors(n);    if (n in factors) {        // n prime        Term[] terms;        foreach (i; 0..n) {            terms ~= Term(1, i);        }        return Polynomial(terms);    } else if (factors.length == 2 && 2 in factors && factors[2] == 1 && (n / 2) in factors && factors[n / 2] == 1) {        //  n = 2p        auto prime = n / 2;        Term[] terms;        auto coeff = -1;        foreach (i; 0..prime) {            coeff *= -1;            terms ~= Term(coeff, i);        }        return Polynomial(terms);    } else if (factors.length == 1 && 2 in factors) {        //  n = 2^h        auto h = factors[2];        Term[] terms;        terms ~= Term(1, 2 ^^ (h - 1));        terms ~= Term(1, 0);        return Polynomial(terms);    } else if (factors.length == 1 && n !in factors) {        // n = p^k        auto p = 0;        auto k = 0;        foreach (prime, v; factors) {            if (prime > p) {                p = prime;                k = v;            }        }        Term[] terms;        foreach (i; 0..p) {            terms ~= Term(1, (i * p) ^^ (k - 1));        }        return Polynomial(terms);    } else if (factors.length == 2 && 2 in factors) {        // n = 2^h * p^k        auto p = 0;        auto k = 0;        foreach (prime, v; factors) {            if (prime != 2 && prime > p) {                p = prime;                k = v;            }        }        Term[] terms;        auto coeff = -1;        auto twoExp = 2 ^^ (factors[2] - 1);        foreach (i; 0..p) {            coeff *= -1;            auto exponent = i * twoExp * p ^^ (k - 1);            terms ~= Term(coeff, exponent);        }        return Polynomial(terms);    } else if (2 in factors && n / 2 % 2 == 1 && n / 2 > 1) {        //  CP(2m)[x] = CP(-m)[x], n odd integer > 1        auto cycloDiv2 = memoize!cyclotomicPolynomialImpl(n / 2);        Term[] terms;        foreach (term; cycloDiv2) {            if (term.exponent() % 2 == 0) {                terms ~= term;            } else {                terms ~= -term;            }        }        return Polynomial(terms);    }     if (ALGORITHM == 0) {        //  Slow - uses basic definition.        auto divisors = getDivisors(n);        //  Polynomial:  ( x^n - 1 )        auto cyclo = Polynomial(Term(1, n), Term(-1, 0));        foreach (i; divisors) {            auto p = memoize!cyclotomicPolynomialImpl(i);            cyclo = cyclo / p;        }        return cyclo;    }    if (ALGORITHM == 1) {        //  Faster.  Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor        auto divisors = getDivisors(n);        auto maxDivisor = int.min;        foreach (div; divisors) {            maxDivisor = max(maxDivisor, div);        }        int[] divisorsExceptMax;        foreach (div; divisors) {            if (maxDivisor % div != 0) {                divisorsExceptMax ~= div;            }        }         //  Polynomial:  ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor        auto cyclo = Polynomial(Term(1, n), Term(-1, 0)) / Polynomial(Term(1, maxDivisor), Term(-1, 0));        foreach (i; divisorsExceptMax) {            auto p = memoize!cyclotomicPolynomialImpl(i);            cyclo = cyclo / p;        }        return cyclo;    }    if (ALGORITHM == 2) {        //  Fastest        //  Let p ; q be primes such that p does not divide n, and q q divides n.        //  Then CP(np)[x] = CP(n)[x^p] / CP(n)[x]        auto m = 1;        auto cyclo = memoize!cyclotomicPolynomialImpl(m);        auto primes = factors.keys;        primes.sort;        foreach (prime; primes) {            //  CP(m)[x]            auto cycloM = cyclo;            //  Compute CP(m)[x^p].            Term[] terms;            foreach (term; cycloM) {                terms ~= Term(term.coefficient(), term.exponent() * prime);            }            cyclo = Polynomial(terms) / cycloM;            m *= prime;        }        //  Now, m is the largest square free divisor of n        auto s = n / m;        //  Compute CP(n)[x] = CP(m)[x^s]        Term[] terms;        foreach (term; cyclo) {            terms ~= Term(term.coefficient(), term.exponent() * s);        }        return Polynomial(terms);    }    assert(false, "Error 103: Invalid algorithm");}alias cyclotomicPolynomial = memoize!cyclotomicPolynomialImpl; void main() {    writeln("Task 1:  cyclotomic polynomials for n <= 30:");   foreach (i; 1 .. 31) {        auto p = cyclotomicPolynomial(i);        writefln("CP[%d] = %s", i, p);   }    writeln;     writeln("Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:");    auto n = 0;    foreach (i; 1 .. 11) {         while (true) {            n++;            auto cyclo = cyclotomicPolynomial(n);            if (cyclo.hasCoefficientAbs(i)) {                writefln("CP[%d] has coefficient with magnitude = %d", n, i);                n--;                break;            }         }    }}`
Output:
```Task 1:  cyclotomic polynomials for n <= 30:
CP[1] = x - 1
CP[2] = x + 1
CP[3] = x^2 + x + 1
CP[4] = x^2 + 1
CP[5] = x^4 + x^3 + x^2 + x + 1
CP[6] = x^2 - x + 1
CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[8] = x^4 + 1
CP[9] = x^6 + x^3 + 1
CP[10] = x^4 - x^3 + x^2 - x + 1
CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[12] = x^4 - x^2 + 1
CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
CP[16] = x^8 + 1
CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[18] = x^6 - x^3 + 1
CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[20] = x^8 - x^6 + x^4 - x^2 + 1
CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[24] = x^8 - x^4 + 1
CP[25] = x^20 + x^15 + x^10 + x^5 + 1
CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[27] = x^36 + x^9 + 1
CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:
CP[1] has coefficient with magnitude = 1
CP[105] has coefficient with magnitude = 2
CP[385] has coefficient with magnitude = 3
CP[1365] has coefficient with magnitude = 4
CP[1785] has coefficient with magnitude = 5
CP[2805] has coefficient with magnitude = 6
CP[3135] has coefficient with magnitude = 7
CP[6545] has coefficient with magnitude = 8
CP[6545] has coefficient with magnitude = 9
CP[10465] has coefficient with magnitude = 10```

## Fermat

This isn't terribly efficient if you have to calculate many cyclotomics- store them in an array rather than using recursion instead if you need to do that- but it showcases Fermat's strength at polynomial expressions.

` &(J=x);                                                    {adjoin x as the variable in the polynomials} Func Cyclotomic(n) =    if n=1 then x-1 fi;                                    {first cyclotomic polynomial is x^n-1}                               r:=x^n-1;                                              {caclulate cyclotomic by division}    for d = 1 to n-1 do        if Divides(d,n) then            r:=r\Cyclotomic(d)        fi;    od;    r.;                                                    {return the polynomial} Func Hascoef(n, k) =    p:=Cyclotomic(n);    for d = 0 to Deg(p) do        if |(Coef(p,d))|=k then Return(1) fi    od;    0.; for d = 1 to 30 do    !!(d,' : ',Cyclotomic(d))od; for m = 1 to 10 do    i:=1;    while not Hascoef(i, m) do        i:+    od;    !!(m,'   :   ',i);od;`

## Go

Translation of: Java
`package main import (    "fmt"    "log"    "math"    "sort"    "strings") const (    algo          = 2    maxAllFactors = 100000) func iabs(i int) int {    if i < 0 {        return -i    }    return i} type term struct{ coef, exp int } func (t term) mul(t2 term) term {    return term{t.coef * t2.coef, t.exp + t2.exp}} func (t term) add(t2 term) term {    if t.exp != t2.exp {        log.Fatal("exponents unequal in term.add method")    }    return term{t.coef + t2.coef, t.exp}} func (t term) negate() term { return term{-t.coef, t.exp} } func (t term) String() string {    switch {    case t.coef == 0:        return "0"    case t.exp == 0:        return fmt.Sprintf("%d", t.coef)    case t.coef == 1:        if t.exp == 1 {            return "x"        } else {            return fmt.Sprintf("x^%d", t.exp)        }    case t.exp == 1:        return fmt.Sprintf("%dx", t.coef)    }    return fmt.Sprintf("%dx^%d", t.coef, t.exp)} type poly struct{ terms []term } // pass coef, exp in pairs as parametersfunc newPoly(values ...int) poly {    le := len(values)    if le == 0 {        return poly{[]term{term{0, 0}}}    }    if le%2 != 0 {        log.Fatalf("odd number of parameters (%d) passed to newPoly function", le)    }    var terms []term    for i := 0; i < le; i += 2 {        terms = append(terms, term{values[i], values[i+1]})    }    p := poly{terms}.tidy()    return p} func (p poly) hasCoefAbs(coef int) bool {    for _, t := range p.terms {        if iabs(t.coef) == coef {            return true        }    }    return false} func (p poly) add(p2 poly) poly {    p3 := newPoly()    le, le2 := len(p.terms), len(p2.terms)    for le > 0 || le2 > 0 {        if le == 0 {            p3.terms = append(p3.terms, p2.terms[le2-1])            le2--        } else if le2 == 0 {            p3.terms = append(p3.terms, p.terms[le-1])            le--        } else {            t := p.terms[le-1]            t2 := p2.terms[le2-1]            if t.exp == t2.exp {                t3 := t.add(t2)                if t3.coef != 0 {                    p3.terms = append(p3.terms, t3)                }                le--                le2--            } else if t.exp < t2.exp {                p3.terms = append(p3.terms, t)                le--            } else {                p3.terms = append(p3.terms, t2)                le2--            }        }    }    return p3.tidy()} func (p poly) addTerm(t term) poly {    q := newPoly()    added := false    for i := 0; i < len(p.terms); i++ {        ct := p.terms[i]        if ct.exp == t.exp {            added = true            if ct.coef+t.coef != 0 {                q.terms = append(q.terms, ct.add(t))            }        } else {            q.terms = append(q.terms, ct)        }    }    if !added {        q.terms = append(q.terms, t)    }    return q.tidy()} func (p poly) mulTerm(t term) poly {    q := newPoly()    for i := 0; i < len(p.terms); i++ {        ct := p.terms[i]        q.terms = append(q.terms, ct.mul(t))    }    return q.tidy()} func (p poly) div(v poly) poly {    q := newPoly()    lcv := v.leadingCoef()    dv := v.degree()    for p.degree() >= v.degree() {        lcp := p.leadingCoef()        s := lcp / lcv        t := term{s, p.degree() - dv}        q = q.addTerm(t)        p = p.add(v.mulTerm(t.negate()))    }    return q.tidy()} func (p poly) leadingCoef() int {    return p.terms[0].coef} func (p poly) degree() int {    return p.terms[0].exp} func (p poly) String() string {    var sb strings.Builder    first := true    for _, t := range p.terms {        if first {            sb.WriteString(t.String())            first = false        } else {            sb.WriteString(" ")            if t.coef > 0 {                sb.WriteString("+ ")                sb.WriteString(t.String())            } else {                sb.WriteString("- ")                sb.WriteString(t.negate().String())            }        }    }    return sb.String()} // in place descending sort by term.expfunc (p poly) sortTerms() {    sort.Slice(p.terms, func(i, j int) bool {        return p.terms[i].exp > p.terms[j].exp    })} // sort terms and remove any unnecesary zero termsfunc (p poly) tidy() poly {    p.sortTerms()    if p.degree() == 0 {        return p    }    for i := len(p.terms) - 1; i >= 0; i-- {        if p.terms[i].coef == 0 {            copy(p.terms[i:], p.terms[i+1:])            p.terms[len(p.terms)-1] = term{0, 0}            p.terms = p.terms[:len(p.terms)-1]        }    }    if len(p.terms) == 0 {        p.terms = append(p.terms, term{0, 0})    }    return p} func getDivisors(n int) []int {    var divs []int    sqrt := int(math.Sqrt(float64(n)))    for i := 1; i <= sqrt; i++ {        if n%i == 0 {            divs = append(divs, i)            d := n / i            if d != i && d != n {                divs = append(divs, d)            }        }    }    return divs} var (    computed   = make(map[int]poly)    allFactors = make(map[int]map[int]int)) func init() {    f := map[int]int{2: 1}    allFactors[2] = f} func getFactors(n int) map[int]int {    if f, ok := allFactors[n]; ok {        return f    }    factors := make(map[int]int)    if n%2 == 0 {        factorsDivTwo := getFactors(n / 2)        for k, v := range factorsDivTwo {            factors[k] = v        }        factors[2]++        if n < maxAllFactors {            allFactors[n] = factors        }        return factors    }    prime := true    sqrt := int(math.Sqrt(float64(n)))    for i := 3; i <= sqrt; i += 2 {        if n%i == 0 {            prime = false            for k, v := range getFactors(n / i) {                factors[k] = v            }            factors[i]++            if n < maxAllFactors {                allFactors[n] = factors            }            return factors        }    }    if prime {        factors[n] = 1        if n < maxAllFactors {            allFactors[n] = factors        }    }    return factors} func cycloPoly(n int) poly {    if p, ok := computed[n]; ok {        return p    }    if n == 1 {        // polynomial: x - 1        p := newPoly(1, 1, -1, 0)        computed[1] = p        return p    }    factors := getFactors(n)    cyclo := newPoly()    if _, ok := factors[n]; ok {        // n is prime        for i := 0; i < n; i++ {            cyclo.terms = append(cyclo.terms, term{1, i})        }    } else if len(factors) == 2 && factors[2] == 1 && factors[n/2] == 1 {        // n == 2p        prime := n / 2        coef := -1        for i := 0; i < prime; i++ {            coef *= -1            cyclo.terms = append(cyclo.terms, term{coef, i})        }    } else if len(factors) == 1 {        if h, ok := factors[2]; ok {            // n == 2^h            cyclo.terms = append(cyclo.terms, term{1, 1 << (h - 1)}, term{1, 0})        } else if _, ok := factors[n]; !ok {            // n == p ^ k            p := 0            for prime := range factors {                p = prime            }            k := factors[p]            for i := 0; i < p; i++ {                pk := int(math.Pow(float64(p), float64(k-1)))                cyclo.terms = append(cyclo.terms, term{1, i * pk})            }        }    } else if len(factors) == 2 && factors[2] != 0 {        // n = 2^h * p^k        p := 0        for prime := range factors {            if prime != 2 {                p = prime            }        }        coef := -1        twoExp := 1 << (factors[2] - 1)        k := factors[p]        for i := 0; i < p; i++ {            coef *= -1            pk := int(math.Pow(float64(p), float64(k-1)))            cyclo.terms = append(cyclo.terms, term{coef, i * twoExp * pk})        }    } else if factors[2] != 0 && ((n/2)%2 == 1) && (n/2) > 1 {        //  CP(2m)[x] == CP(-m)[x], n odd integer > 1        cycloDiv2 := cycloPoly(n / 2)        for _, t := range cycloDiv2.terms {            t2 := t            if t.exp%2 != 0 {                t2 = t.negate()            }            cyclo.terms = append(cyclo.terms, t2)        }    } else if algo == 0 {        // slow - uses basic definition        divs := getDivisors(n)        // polynomial: x^n - 1        cyclo = newPoly(1, n, -1, 0)        for _, i := range divs {            p := cycloPoly(i)            cyclo = cyclo.div(p)        }    } else if algo == 1 {        //  faster - remove max divisor (and all divisors of max divisor)        //  only one divide for all divisors of max divisor        divs := getDivisors(n)        maxDiv := math.MinInt32        for _, d := range divs {            if d > maxDiv {                maxDiv = d            }        }        var divsExceptMax []int        for _, d := range divs {            if maxDiv%d != 0 {                divsExceptMax = append(divsExceptMax, d)            }        }        // polynomial:  ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor        cyclo = newPoly(1, n, -1, 0)        cyclo = cyclo.div(newPoly(1, maxDiv, -1, 0))        for _, i := range divsExceptMax {            p := cycloPoly(i)            cyclo = cyclo.div(p)        }    } else if algo == 2 {        //  fastest        //  let p, q be primes such that p does not divide n, and q divides n        //  then CP(np)[x] = CP(n)[x^p] / CP(n)[x]        m := 1        cyclo = cycloPoly(m)        var primes []int        for prime := range factors {            primes = append(primes, prime)        }        sort.Ints(primes)        for _, prime := range primes {            //  CP(m)[x]            cycloM := cyclo            //  compute CP(m)[x^p]            var terms []term            for _, t := range cycloM.terms {                terms = append(terms, term{t.coef, t.exp * prime})            }            cyclo = newPoly()            cyclo.terms = append(cyclo.terms, terms...)            cyclo = cyclo.tidy()            cyclo = cyclo.div(cycloM)            m *= prime        }        //  now, m is the largest square free divisor of n        s := n / m        //  Compute CP(n)[x] = CP(m)[x^s]        var terms []term        for _, t := range cyclo.terms {            terms = append(terms, term{t.coef, t.exp * s})        }        cyclo = newPoly()        cyclo.terms = append(cyclo.terms, terms...)    } else {        log.Fatal("invalid algorithm")    }    cyclo = cyclo.tidy()    computed[n] = cyclo    return cyclo} func main() {    fmt.Println("Task 1:  cyclotomic polynomials for n <= 30:")    for i := 1; i <= 30; i++ {        p := cycloPoly(i)        fmt.Printf("CP[%2d] = %s\n", i, p)    }     fmt.Println("\nTask 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:")    n := 0    for i := 1; i <= 10; i++ {        for {            n++            cyclo := cycloPoly(n)            if cyclo.hasCoefAbs(i) {                fmt.Printf("CP[%d] has coefficient with magnitude = %d\n", n, i)                n--                break            }        }    }}`
Output:
```Task 1:  cyclotomic polynomials for n <= 30:
CP[ 1] = x - 1
CP[ 2] = x + 1
CP[ 3] = x^2 + x + 1
CP[ 4] = x^2 + 1
CP[ 5] = x^4 + x^3 + x^2 + x + 1
CP[ 6] = x^2 - x + 1
CP[ 7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[ 8] = x^4 + 1
CP[ 9] = x^6 + x^3 + 1
CP[10] = x^4 - x^3 + x^2 - x + 1
CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[12] = x^4 - x^2 + 1
CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
CP[16] = x^8 + 1
CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[18] = x^6 - x^3 + 1
CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[20] = x^8 - x^6 + x^4 - x^2 + 1
CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[24] = x^8 - x^4 + 1
CP[25] = x^20 + x^15 + x^10 + x^5 + 1
CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[27] = x^18 + x^9 + 1
CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:
CP[1] has coefficient with magnitude = 1
CP[105] has coefficient with magnitude = 2
CP[385] has coefficient with magnitude = 3
CP[1365] has coefficient with magnitude = 4
CP[1785] has coefficient with magnitude = 5
CP[2805] has coefficient with magnitude = 6
CP[3135] has coefficient with magnitude = 7
CP[6545] has coefficient with magnitude = 8
CP[6545] has coefficient with magnitude = 9
CP[10465] has coefficient with magnitude = 10
```

${\displaystyle Insertformulahere}$

Uses synthetic polynomial division and simple memoization.

`import Data.Listimport Data.Numbers.Primes (primeFactors) negateVar p = zipWith (*) p \$ reverse \$ take (length p) \$ cycle [1,-1] lift p 1 = plift p n = intercalate (replicate (n-1) 0) (pure <\$> p) shortDiv :: [Integer] -> [Integer] -> [Integer]shortDiv p1 (_:p2) = unfoldr go (length p1 - length p2, p1)  where    go (0, _) = Nothing    go (i, h:t) = Just (h, (i-1, zipWith (+) (map (h *) ker) t))    ker = negate <\$> p2 ++ repeat 0 primePowerFactors = sortOn fst . map (\x-> (head x, length x)) . group . primeFactors -- simple memoizationcyclotomics :: [[Integer]]cyclotomics = cyclotomic <\$> [0..] cyclotomic :: Int -> [Integer]cyclotomic 0 = [0]cyclotomic 1 = [1, -1]cyclotomic 2 = [1, 1]cyclotomic n = case primePowerFactors n of  -- for n = 2^k  [(2,h)]       -> 1 : replicate (2 ^ (h-1) - 1) 0 ++ [1]  -- for prime n  [(p,1)]       -> replicate n 1  -- for power of prime n  [(p,m)]       -> lift (cyclotomics !! p) (p^(m-1))  -- for n = 2*p and prime p  [(2,1),(p,1)] -> take (n `div` 2) \$ cycle [1,-1]  -- for n = 2*m and odd m  (2,1):_       -> negateVar \$ cyclotomics !! (n `div` 2)  -- general case  (p, m):ps     -> let cm = cyclotomics !! (n `div` (p ^ m))                   in lift (lift cm p `shortDiv` cm) (p^(m-1))`

Simple examples

```λ> cyclotomic 7
[1,1,1,1,1,1,1]

λ> cyclotomic 9
[1,0,0,1,0,0,1]

λ> cyclotomic 16
[1,0,0,0,0,0,0,0,1]```

`showPoly [] = "0"showPoly p = foldl1 (\r -> (r ++) . term) \$             dropWhile null \$             foldMap (\(c, n) -> [show c ++ expt n]) \$             zip (reverse p) [0..]  where    expt = \case 0 -> ""                 1 -> "*x"                 n -> "*x^" ++ show n     term = \case [] -> ""                 '0':'*':t -> ""                 '-':'1':'*':t -> " - " ++ t                 '1':'*':t -> " + " ++ t                 '-':t -> " - " ++ t                 t -> " + " ++ t      main = do  mapM_ (print . showPoly . cyclotomic) [1..30]  putStrLn \$ replicate 40 '-'   mapM_ showLine \$ take 4 task2  where    showLine (j, i, l) = putStrLn \$ concat [ show j                                            , " appears in CM(", show i                                            , ") of length ", show l ]     -- in order to make computations faster we leave only each 5-th polynomial    task2 = (1,1,2) : tail (search 1 \$ zip [0,5..] \$ skipBy 5 cyclotomics)      where        search i ((k, p):ps) = if i `notElem` (abs <\$> p)                               then search i ps                               else (i, k, length p) : search (i+1) ((k, p):ps) skipBy n [] = []skipBy n lst = let (x:_, b) = splitAt n lst in x:skipBy n b`

Result

```"-1 + x^1"
"1 + x^1"
"1 + x^1 + x^2"
"1 + x^2"
"1 + x^1 + x^2 + x^3 + x^4"
"1 - x^1 + x^2"
"1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6"
"1 + x^4"
"1 + x^3 + x^6"
"1 - x^1 + x^2 - x^3 + x^4"
"1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10"
"1 - x^2 + x^4"
"1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12"
"1 - x^1 + x^2 - x^3 + x^4 - x^5 + x^6"
"1 - x^1 + x^3 - x^4 + x^5 - x^7 + x^8"
"1 + x^8"
"1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16"
"1 - x^3 + x^6"
"1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18"
"1 - x^2 + x^4 - x^6 + x^8"
"1 - x^1 + x^3 - x^4 + x^6 - x^8 + x^9 - x^11 + x^12"
"1 - x^1 + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10"
"1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + x^20 + x^21 + x^22"
"1 - x^4 + x^8"
"1 + x^5 + x^10 + x^15 + x^20"
"1 - x^1 + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12"
"1 + x^9 + x^18"
"1 - x^2 + x^4 - x^6 + x^8 - x^10 + x^12"
"1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + x^20 + x^21 + x^22 + x^23 + x^24 + x^25 + x^26 + x^27 + x^28"
"1 + x^1 - x^3 - x^4 - x^5 + x^7 + x^8"
----------------------------------------
1 appears in CM(1) having 2 terms
2 appears in CM(105) having 49 terms
3 appears in CM(385) having 241 terms
4 appears in CM(1365) having 577 terms
5 appears in CM(1785) having 769 terms
6 appears in CM(2805) having 1281 terms
7 appears in CM(3135) having 1441 terms
8 appears in CP(6545) having 3841 terms
9 appears in CP(6545) having 3841 terms
10 appears in CP(10465) having 6337 terms```

Computations take a while...

## J

For values up to 70, we can find cyclotomic polynomials by finding a polynomial with roots of unity relatively prime to the order of the polynomial:

`cyclo=: {{<.-:1+(++) p. 1;^0j2p1* y%~1+I.1=y+.1+i.y}}`

This approach suggests that cyclotomic polynomial zero should be f0(x)= 1

Routine to find the nth cyclotomic polynomial:

`{{ if.0>nc<'cache' do.cache=:y end.}} (,1);_1 1 cyclotomic=: {{   if.y<#cache do.     if.#c=. y{::cache do.       c return.     end.   end.   c=. unpad cyclotomic000 y   if. y>:#cache do. cache=:(100+y){.cache end.   cache=: (<c) y} cache   c}} cyclotomic000=:  {{ assert.0<y   'q p'=. __ q: y   if. 1=#q do.     ,(y%*/q) {."0 q#1   elseif.2={.q do.     ,(y%*/q) {."0 (* 1 _1 \$~ #) cyclotomic */}.q   elseif. 1 e. 1 < p do.     ,(y%*/q) {."0 cyclotomic */q   else.     (_1,(-y){.1) pDiv ;+//[email protected](*/)each/ cyclotomic each}:*/@>,{1,each q   end.}}  NB. discard high order zero coefficients in representation of polynomialunpad=: {.~ 1+0 i:~0=] NB. polynomial division, optimized for somewhat sparse polynomialspDiv=: {{  q=. \$j=. 2 + x -&# y  'x y'=. x,:y  while. j=. j-1 do.    if. 0={.x do. j=. j-<:i=. 0 i.~ 0=x      q=. q,i#0      x=. i |.!.0 x    else.      q=. q, r=. x %&{. y      x=. 1 |.!.0 x - y*r    end.  end.q}}`

If you take all the divisors of a number. (For example, for 12, the divisors are: 1, 2, 3, 4, 6 and 12) and find the product of their cyclotomic polynomials (for example, for 12, x-1, x+1, x2+x+1, x2+1, x2-x+1, and x4-x2+1) you get xn-1 (for 12, that would of course be x12-1).

Notes:

• the coefficients of cyclotomic polynomials after 1 form a palindrome (that's the q#1 phrase in the implementation).
• the cyclotomic polynomial for a prime number has as many terms as that number, and the coefficients are all 1 (with no intervening zeros -- the highest power is one less than that prime).
• powers of primes add zero coefficients to the polynomial (that's the ,(y%*/q) {."0 ... phrase in the implementation). This means that we can mostly ignore powers of prime numbers -- they're just going to correspond to zeros we add to the base polynomial.
• an even base cyclotomic polynomial is the same as the corresponding odd base cyclotomic polynomial except with x replaced by negative x. (that's the (* 1 _1 \$~ #) phrase in the implementation.
• To deal with the general case, we use polynomial division, xn-1 divided by the polynomial product the cyclotomic polynomials of the proper divisors of number we're looking for.
• +//[email protected](*/) is polynomial product in J.

`taskfmt=: {{  c=. ":each j=.cyclotomic y  raw=. rplc&'_-' ;:inv}.,'+';"0|.(*|j)#c('(',[,],')'"_)each '*x^',&":L:0 <"0 i.#c  txt=. raw rplc'(1*x^0)';'1';'(-1*x^0)';'(-1)';'*x^1)';'*x)'  LF,~'CP[',y,&":']= ',rplc&('(x)';'x';'+ (-1)';'- 1')txt rplc'(1*';'(';'(-1*';'(-'}} taskorder=: {{  r=.\$k=.0  while.y>#r do.k=.k+1    if.(1+#r) e.|cyclotomic k do.      r=. r,k      k=. k-1    end.  end.r}}    ;taskfmt each 1+i.30CP[1]= x - 1CP[2]= x + 1CP[3]= (x^2) + x + 1CP[4]= (x^2) + 1CP[5]= (x^4) + (x^3) + (x^2) + x + 1CP[6]= (x^2) + (-x) + 1CP[7]= (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1CP[8]= (x^4) + 1CP[9]= (x^6) + (x^3) + 1CP[10]= (x^4) + (-x^3) + (x^2) + (-x) + 1CP[11]= (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1CP[12]= (x^4) + (-x^2) + 1CP[13]= (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1CP[14]= (x^6) + (-x^5) + (x^4) + (-x^3) + (x^2) + (-x) + 1CP[15]= (x^8) + (-x^7) + (x^5) + (-x^4) + (x^3) + (-x) + 1CP[16]= (x^8) + 1CP[17]= (x^16) + (x^15) + (x^14) + (x^13) + (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1CP[18]= (x^6) + (-x^3) + 1CP[19]= (x^18) + (x^17) + (x^16) + (x^15) + (x^14) + (x^13) + (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1CP[20]= (x^8) + (-x^6) + (x^4) + (-x^2) + 1CP[21]= (x^12) + (-x^11) + (x^9) + (-x^8) + (x^6) + (-x^4) + (x^3) + (-x) + 1CP[22]= (x^10) + (-x^9) + (x^8) + (-x^7) + (x^6) + (-x^5) + (x^4) + (-x^3) + (x^2) + (-x) + 1CP[23]= (x^22) + (x^21) + (x^20) + (x^19) + (x^18) + (x^17) + (x^16) + (x^15) + (x^14) + (x^13) + (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1CP[24]= (x^8) + (-x^4) + 1CP[25]= (x^20) + (x^15) + (x^10) + (x^5) + 1CP[26]= (x^12) + (-x^11) + (x^10) + (-x^9) + (x^8) + (-x^7) + (x^6) + (-x^5) + (x^4) + (-x^3) + (x^2) + (-x) + 1CP[27]= (x^18) + (x^9) + 1CP[28]= (x^12) + (-x^10) + (x^8) + (-x^6) + (x^4) + (-x^2) + 1CP[29]= (x^28) + (x^27) + (x^26) + (x^25) + (x^24) + (x^23) + (x^22) + (x^21) + (x^20) + (x^19) + (x^18) + (x^17) + (x^16) + (x^15) + (x^14) + (x^13) + (x^12) + (x^11) + (x^10) + (x^9) + (x^8) + (x^7) + (x^6) + (x^5) + (x^4) + (x^3) + (x^2) + x + 1CP[30]= (x^8) + (x^7) + (-x^5) + (-x^4) + (-x^3) + x + 1    (,.~#\) taskorder 10 1     1 2   105 3   385 4  1365 5  1785 6  2805 7  3135 8  6545 9  654510 10465`

### Another approach

As noted in the J programming forum, we can improve the big-O character of this algorithm by using the fast fourier transform for polynomial multiplication and division.

`NB. install'math/fftw'require'math/fftw' cyclotomic000=:  {{ assert.0<y  if. y = 1 do. _1 1 return. end.  'q p'=. __ q: y  if. 1=#q do.    ,(y%*/q) {."0 q#1  elseif.2={.q do.    ,(y%*/q) {."0 (* 1 _1 \$~ #) cyclotomic */}.q  elseif. 1 e. 1 < p do.    ,(y%*/q) {."0 cyclotomic */q  else.    lgl=. {:\$ ctlist=. cyclotomic "0 }:*/@>,{1,each q    NB. ctlist is 2-d table of polynomial divisors    lgd=. # dividend=. _1,(-y){.1                        NB. (x^n) - 1, and its size    lg=.  >.&.(2&^.)  lgl >. lgd                         NB. required lengths of all polynomials for fft transforms                      NB. really, "divisor" is the fft of the divisor!            divisor=. */ fftw"1 lg{."1 ctlist            NB. FFT article doesn't deal with lists of multiplicands    unpad roundreal ifftw"1 divisor %~ fftw lg{.dividend NB. similar to article's multiplication  end.}} roundreal =: [: <. 0.5 + 9&o.`

This variation for polynomial division is only valid when there's no remainder to be concerned with (which is the case, here). The article mentioned in the comments is an essay on using fft for polynomial multiplication

This approach gave slightly over a 16x speedup for taskorder 10, from a 2 element cache, with an approximately 50% increased memory footprint. (Remember, of course, that benchmarks and benchmark ratios have dependencies on computer architecture and language implementation, and the host environment.)

## Java

` import java.util.ArrayList;import java.util.Collections;import java.util.Comparator;import java.util.HashMap;import java.util.List;import java.util.Map;import java.util.TreeMap; public class CyclotomicPolynomial {     @SuppressWarnings("unused")    private static int divisions = 0;    private static int algorithm = 2;     public static void main(String[] args) throws Exception {        System.out.println("Task 1:  cyclotomic polynomials for n <= 30:");        for ( int i = 1 ; i <= 30 ; i++ ) {            Polynomial p = cyclotomicPolynomial(i);            System.out.printf("CP[%d] = %s%n", i, p);        }        System.out.println("Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:");        int n = 0;        for ( int i = 1 ; i <= 10 ; i++ ) {            while ( true ) {                n++;                Polynomial cyclo = cyclotomicPolynomial(n);                if ( cyclo.hasCoefficientAbs(i) ) {                    System.out.printf("CP[%d] has coefficient with magnitude = %d%n", n, i);                    n--;                    break;                }            }        }    }     private static final Map<Integer, Polynomial> COMPUTED = new HashMap<>();     private static Polynomial cyclotomicPolynomial(int n) {        if ( COMPUTED.containsKey(n) ) {            return COMPUTED.get(n);        }         //System.out.println("COMPUTE:  n = " + n);         if ( n == 1 ) {            //  Polynomial:  x - 1            Polynomial p = new Polynomial(1, 1, -1, 0);            COMPUTED.put(1, p);            return p;        }         Map<Integer,Integer> factors = getFactors(n);         if ( factors.containsKey(n) ) {            //  n prime            List<Term> termList = new ArrayList<>();            for ( int index = 0 ; index < n ; index++ ) {                termList.add(new Term(1, index));            }             Polynomial cyclo = new Polynomial(termList);            COMPUTED.put(n, cyclo);            return cyclo;        }        else if ( factors.size() == 2 && factors.containsKey(2) && factors.get(2) == 1 && factors.containsKey(n/2) && factors.get(n/2) == 1 ) {            //  n = 2p            int prime = n/2;            List<Term> termList = new ArrayList<>();            int coeff = -1;            for ( int index = 0 ; index < prime ; index++ ) {                coeff *= -1;                termList.add(new Term(coeff, index));            }             Polynomial cyclo = new Polynomial(termList);            COMPUTED.put(n, cyclo);            return cyclo;        }        else if ( factors.size() == 1 && factors.containsKey(2) ) {            //  n = 2^h            int h = factors.get(2);            List<Term> termList = new ArrayList<>();            termList.add(new Term(1, (int) Math.pow(2, h-1)));            termList.add(new Term(1, 0));            Polynomial cyclo = new Polynomial(termList);            COMPUTED.put(n, cyclo);            return cyclo;        }        else if ( factors.size() == 1 && ! factors.containsKey(n) ) {            // n = p^k            int p = 0;            for ( int prime : factors.keySet() ) {                p = prime;            }            int k = factors.get(p);            List<Term> termList = new ArrayList<>();            for ( int index = 0 ; index < p ; index++ ) {                termList.add(new Term(1, index * (int) Math.pow(p, k-1)));            }             Polynomial cyclo = new Polynomial(termList);            COMPUTED.put(n, cyclo);            return cyclo;        }        else if ( factors.size() == 2 && factors.containsKey(2) ) {            //  n = 2^h * p^k            int p = 0;            for ( int prime : factors.keySet() ) {                if ( prime != 2 ) {                    p = prime;                }            }            List<Term> termList = new ArrayList<>();            int coeff = -1;            int twoExp = (int) Math.pow(2, factors.get(2)-1);            int k = factors.get(p);            for ( int index = 0 ; index < p ; index++ ) {                coeff *= -1;                termList.add(new Term(coeff, index * twoExp * (int) Math.pow(p, k-1)));            }             Polynomial cyclo = new Polynomial(termList);            COMPUTED.put(n, cyclo);            return cyclo;                    }        else if ( factors.containsKey(2) && ((n/2) % 2 == 1) && (n/2) > 1 ) {            //  CP(2m)[x] = CP(-m)[x], n odd integer > 1            Polynomial cycloDiv2 = cyclotomicPolynomial(n/2);            List<Term> termList = new ArrayList<>();            for ( Term term : cycloDiv2.polynomialTerms ) {                termList.add(term.exponent % 2 == 0 ? term : term.negate());            }            Polynomial cyclo = new Polynomial(termList);            COMPUTED.put(n, cyclo);            return cyclo;                    }         //  General Case         if ( algorithm == 0 ) {            //  Slow - uses basic definition.            List<Integer> divisors = getDivisors(n);            //  Polynomial:  ( x^n - 1 )            Polynomial cyclo = new Polynomial(1, n, -1, 0);            for ( int i : divisors ) {                Polynomial p = cyclotomicPolynomial(i);                cyclo = cyclo.divide(p);            }             COMPUTED.put(n, cyclo);                        return cyclo;        }        else if ( algorithm == 1 ) {            //  Faster.  Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor            List<Integer> divisors = getDivisors(n);            int maxDivisor = Integer.MIN_VALUE;            for ( int div : divisors ) {                maxDivisor = Math.max(maxDivisor, div);            }            List<Integer> divisorsExceptMax = new ArrayList<Integer>();            for ( int div : divisors ) {                if ( maxDivisor % div != 0 ) {                    divisorsExceptMax.add(div);                }            }             //  Polynomial:  ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor            Polynomial cyclo = new Polynomial(1, n, -1, 0).divide(new Polynomial(1, maxDivisor, -1, 0));            for ( int i : divisorsExceptMax ) {                Polynomial p = cyclotomicPolynomial(i);                cyclo = cyclo.divide(p);            }             COMPUTED.put(n, cyclo);             return cyclo;        }        else if ( algorithm == 2 ) {            //  Fastest            //  Let p ; q be primes such that p does not divide n, and q q divides n.            //  Then CP(np)[x] = CP(n)[x^p] / CP(n)[x]            int m = 1;            Polynomial cyclo = cyclotomicPolynomial(m);            List<Integer> primes = new ArrayList<>(factors.keySet());            Collections.sort(primes);            for ( int prime : primes ) {                //  CP(m)[x]                Polynomial cycloM = cyclo;                //  Compute CP(m)[x^p].                List<Term> termList = new ArrayList<>();                for ( Term t : cycloM.polynomialTerms ) {                    termList.add(new Term(t.coefficient, t.exponent * prime));                }                cyclo = new Polynomial(termList).divide(cycloM);                m = m * prime;            }            //  Now, m is the largest square free divisor of n            int s = n / m;            //  Compute CP(n)[x] = CP(m)[x^s]            List<Term> termList = new ArrayList<>();            for ( Term t : cyclo.polynomialTerms ) {                termList.add(new Term(t.coefficient, t.exponent * s));            }            cyclo = new Polynomial(termList);            COMPUTED.put(n, cyclo);             return cyclo;        }        else {            throw new RuntimeException("ERROR 103:  Invalid algorithm.");        }    }     private static final List<Integer> getDivisors(int number) {        List<Integer> divisors = new ArrayList<Integer>();        long sqrt = (long) Math.sqrt(number);        for ( int i = 1 ; i <= sqrt ; i++ ) {            if ( number % i == 0 ) {                divisors.add(i);                int div = number / i;                if ( div != i && div != number ) {                    divisors.add(div);                }            }        }        return divisors;    }     private static final Map<Integer,Map<Integer,Integer>> allFactors = new TreeMap<Integer,Map<Integer,Integer>>();    static {        Map<Integer,Integer> factors = new TreeMap<Integer,Integer>();        factors.put(2, 1);        allFactors.put(2, factors);    }     public static Integer MAX_ALL_FACTORS = 100000;     public static final Map<Integer,Integer> getFactors(Integer number) {        if ( allFactors.containsKey(number) ) {            return allFactors.get(number);        }        Map<Integer,Integer> factors = new TreeMap<Integer,Integer>();        if ( number % 2 == 0 ) {            Map<Integer,Integer> factorsdDivTwo = getFactors(number/2);            factors.putAll(factorsdDivTwo);            factors.merge(2, 1, (v1, v2) -> v1 + v2);            if ( number < MAX_ALL_FACTORS )                 allFactors.put(number, factors);            return factors;        }        boolean prime = true;        long sqrt = (long) Math.sqrt(number);        for ( int i = 3 ; i <= sqrt ; i += 2 ) {            if ( number % i == 0 ) {                prime = false;                factors.putAll(getFactors(number/i));                factors.merge(i, 1, (v1, v2) -> v1 + v2);                if ( number < MAX_ALL_FACTORS )                     allFactors.put(number, factors);                return factors;            }        }        if ( prime ) {            factors.put(number, 1);            if ( number < MAX_ALL_FACTORS )                 allFactors.put(number, factors);        }        return factors;    }     private static final class Polynomial {         private List<Term> polynomialTerms;         //  Format - coeff, exp, coeff, exp, (repeating in pairs) . . .        public Polynomial(int ... values) {            if ( values.length % 2 != 0 ) {                throw new IllegalArgumentException("ERROR 102:  Polynomial constructor.  Length must be even.  Length = " + values.length);            }            polynomialTerms = new ArrayList<>();            for ( int i = 0 ; i < values.length ; i += 2 ) {                Term t = new Term(values[i], values[i+1]);                polynomialTerms.add(t);            }            Collections.sort(polynomialTerms, new TermSorter());        }         public Polynomial() {            //  zero            polynomialTerms = new ArrayList<>();            polynomialTerms.add(new Term(0,0));        }         private boolean hasCoefficientAbs(int coeff) {            for ( Term term : polynomialTerms ) {                if ( Math.abs(term.coefficient) == coeff ) {                    return true;                }            }            return false;        }         private Polynomial(List<Term> termList) {            if ( termList.size() == 0 ) {                //  zero                termList.add(new Term(0,0));            }            else {                //  Remove zero terms if needed                for ( int i = 0 ; i < termList.size() ; i++ ) {                    if ( termList.get(i).coefficient == 0 ) {                        termList.remove(i);                    }                }            }            if ( termList.size() == 0 ) {                //  zero                termList.add(new Term(0,0));            }            polynomialTerms = termList;            Collections.sort(polynomialTerms, new TermSorter());        }         public Polynomial divide(Polynomial v) {            //long start = System.currentTimeMillis();            divisions++;            Polynomial q = new Polynomial();            Polynomial r = this;            long lcv = v.leadingCoefficient();            long dv = v.degree();            while ( r.degree() >= v.degree() ) {                long lcr = r.leadingCoefficient();                long s = lcr / lcv;  //  Integer division                Term term = new Term(s, r.degree() - dv);                q = q.add(term);                r = r.add(v.multiply(term.negate()));            }            //long end = System.currentTimeMillis();            //System.out.printf("Divide:  Elapsed = %d, Degree 1 = %d, Degree 2 = %d%n", (end-start), this.degree(), v.degree());            return q;        }         public Polynomial add(Polynomial polynomial) {            List<Term> termList = new ArrayList<>();            int thisCount = polynomialTerms.size();            int polyCount = polynomial.polynomialTerms.size();            while ( thisCount > 0 || polyCount > 0 ) {                Term thisTerm = thisCount == 0 ? null : polynomialTerms.get(thisCount-1);                Term polyTerm = polyCount == 0 ? null : polynomial.polynomialTerms.get(polyCount-1);                if ( thisTerm == null ) {                    termList.add(polyTerm.clone());                    polyCount--;                }                else if (polyTerm == null ) {                    termList.add(thisTerm.clone());                    thisCount--;                }                else if ( thisTerm.degree() == polyTerm.degree() ) {                    Term t = thisTerm.add(polyTerm);                    if ( t.coefficient != 0 ) {                        termList.add(t);                    }                    thisCount--;                    polyCount--;                }                else if ( thisTerm.degree() < polyTerm.degree() ) {                    termList.add(thisTerm.clone());                    thisCount--;                }                else {                    termList.add(polyTerm.clone());                    polyCount--;                }            }            return new Polynomial(termList);        }         public Polynomial add(Term term) {            List<Term> termList = new ArrayList<>();            boolean added = false;            for ( int index = 0 ; index < polynomialTerms.size() ; index++ ) {                Term currentTerm = polynomialTerms.get(index);                if ( currentTerm.exponent == term.exponent ) {                    added = true;                    if ( currentTerm.coefficient + term.coefficient != 0 ) {                        termList.add(currentTerm.add(term));                    }                }                else {                    termList.add(currentTerm.clone());                }            }            if ( ! added ) {                termList.add(term.clone());            }            return new Polynomial(termList);        }         public Polynomial multiply(Term term) {            List<Term> termList = new ArrayList<>();            for ( int index = 0 ; index < polynomialTerms.size() ; index++ ) {                Term currentTerm = polynomialTerms.get(index);                termList.add(currentTerm.clone().multiply(term));            }            return new Polynomial(termList);        }         public Polynomial clone() {            List<Term> clone = new ArrayList<>();            for ( Term t : polynomialTerms ) {                clone.add(new Term(t.coefficient, t.exponent));            }            return new Polynomial(clone);        }         public long leadingCoefficient() {//            long coefficient = 0;//            long degree = Integer.MIN_VALUE;//            for ( Term t : polynomialTerms ) {//                if ( t.degree() > degree ) {//                    coefficient = t.coefficient;//                    degree = t.degree();//                }//            }            return polynomialTerms.get(0).coefficient;        }         public long degree() {//            long degree = Integer.MIN_VALUE;//            for ( Term t : polynomialTerms ) {//                if ( t.degree() > degree ) {//                    degree = t.degree();//                }//            }            return polynomialTerms.get(0).exponent;        }         @Override        public String toString() {            StringBuilder sb = new StringBuilder();            boolean first = true;            for ( Term term : polynomialTerms ) {                if ( first ) {                    sb.append(term);                    first = false;                }                else {                    sb.append(" ");                    if ( term.coefficient > 0 ) {                        sb.append("+ ");                        sb.append(term);                    }                    else {                        sb.append("- ");                        sb.append(term.negate());                    }                }            }            return sb.toString();        }    }     private static final class TermSorter implements Comparator<Term> {        @Override        public int compare(Term o1, Term o2) {            return (int) (o2.exponent - o1.exponent);        }            }     //  Note:  Cyclotomic Polynomials have small coefficients.  Not appropriate for general polynomial usage.    private static final class Term {        long coefficient;        long exponent;         public Term(long c, long e) {            coefficient = c;            exponent = e;        }         public Term clone() {            return new Term(coefficient, exponent);        }         public Term multiply(Term term) {            return new Term(coefficient * term.coefficient, exponent + term.exponent);        }         public Term add(Term term) {            if ( exponent != term.exponent ) {                throw new RuntimeException("ERROR 102:  Exponents not equal.");            }            return new Term(coefficient + term.coefficient, exponent);        }         public Term negate() {            return new Term(-coefficient, exponent);        }         public long degree() {            return exponent;        }         @Override        public String toString() {            if ( coefficient == 0 ) {                return "0";            }            if ( exponent == 0 ) {                return "" + coefficient;            }            if ( coefficient == 1 ) {                if ( exponent == 1 ) {                    return "x";                }                else {                    return "x^" + exponent;                }            }            if ( exponent == 1 ) {                return coefficient + "x";            }            return coefficient + "x^" + exponent;        }    } } `
Output:
```Task 1:  cyclotomic polynomials for n <= 30:
CP[1] = x - 1
CP[2] = x + 1
CP[3] = x^2 + x + 1
CP[4] = x^2 + 1
CP[5] = x^4 + x^3 + x^2 + x + 1
CP[6] = x^2 - x + 1
CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[8] = x^4 + 1
CP[9] = x^6 + x^3 + 1
CP[10] = x^4 - x^3 + x^2 - x + 1
CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[12] = x^4 - x^2 + 1
CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
CP[16] = x^8 + 1
CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[18] = x^6 - x^3 + 1
CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[20] = x^8 - x^6 + x^4 - x^2 + 1
CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[24] = x^8 - x^4 + 1
CP[25] = x^20 + x^15 + x^10 + x^5 + 1
CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[27] = x^18 + x^9 + 1
CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:
CP[1] has coefficient with magnitude = 1
CP[105] has coefficient with magnitude = 2
CP[385] has coefficient with magnitude = 3
CP[1365] has coefficient with magnitude = 4
CP[1785] has coefficient with magnitude = 5
CP[2805] has coefficient with magnitude = 6
CP[3135] has coefficient with magnitude = 7
CP[6545] has coefficient with magnitude = 8
CP[6545] has coefficient with magnitude = 9
CP[10465] has coefficient with magnitude = 10
```

## Julia

`using Primes, Polynomials # memoize cache for recursive callsconst cyclotomics = Dict([1 => Poly([big"-1", big"1"]), 2 => Poly([big"1", big"1"])]) # get all integer divisors of an integer, except itselffunction divisors(n::Integer)    f = [one(n)]    for (p, e) in factor(n)        f = reduce(vcat, [f * p^j for j in 1:e], init=f)    end    return resize!(f, length(f) - 1)end """    cyclotomic(n::Integer) Calculate the n -th cyclotomic polynomial.See wikipedia article at bottom of section /wiki/Cyclotomic_polynomial#Fundamental_toolsThe algorithm is reliable but slow for large n > 1000."""function cyclotomic(n::Integer)    if haskey(cyclotomics, n)        c = cyclotomics[n]    elseif isprime(n)        c = Poly(ones(BigInt, n))        cyclotomics[n] = c    else  # recursive formula seen in wikipedia article        c = Poly([big"-1"; zeros(BigInt, n - 1); big"1"])        for d in divisors(n)            c ÷= cyclotomic(d)        end        cyclotomics[n] = c    end    return cend println("First 30 cyclotomic polynomials:")for i in 1:30    println(rpad("\$i:  ", 5), cyclotomic(BigInt(i)))end const dig = zeros(BigInt, 10)for i in 1:1000000    if all(x -> x != 0, dig)        break    end    for coef in coeffs(cyclotomic(i))        x = abs(coef)        if 0 < x < 11 && dig[Int(x)] == 0            dig[Int(x)] = coef < 0 ? -i : i        end    endendfor (i, n) in enumerate(dig)    println("The cyclotomic polynomial Φ(", abs(n), ") has a coefficient that is ", n < 0 ? -i : i)end `
Output:
```First 30 cyclotomic polynomials:
1:   Poly(-1 + x)
2:   Poly(1 + x)
3:   Poly(1 + x + x^2)
4:   Poly(1.0 + 1.0*x^2)
5:   Poly(1 + x + x^2 + x^3 + x^4)
6:   Poly(1.0 - 1.0*x + 1.0*x^2)
7:   Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)
8:   Poly(1.0 + 1.0*x^4)
9:   Poly(1.0 + 1.0*x^3 + 1.0*x^6)
10:  Poly(1.0 - 1.0*x + 1.0*x^2 - 1.0*x^3 + 1.0*x^4)
11:  Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10)
12:  Poly(1.0 - 1.0*x^2 + 1.0*x^4)
13:  Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12)
14:  Poly(1.0 - 1.0*x + 1.0*x^2 - 1.0*x^3 + 1.0*x^4 - 1.0*x^5 + 1.0*x^6)
15:  Poly(1.0 - 1.0*x + 1.0*x^3 - 1.0*x^4 + 1.0*x^5 - 1.0*x^7 + 1.0*x^8)
16:  Poly(1.0 + 1.0*x^8)
17:  Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16)
18:  Poly(1.0 - 1.0*x^3 + 1.0*x^6)
19:  Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18)
20:  Poly(1.0 - 1.0*x^2 + 1.0*x^4 - 1.0*x^6 + 1.0*x^8)
21:  Poly(1.0 - 1.0*x + 1.0*x^3 - 1.0*x^4 + 1.0*x^6 - 1.0*x^8 + 1.0*x^9 - 1.0*x^11 + 1.0*x^12)
22:  Poly(1.0 - 1.0*x + 1.0*x^2 - 1.0*x^3 + 1.0*x^4 - 1.0*x^5 + 1.0*x^6 - 1.0*x^7 + 1.0*x^8 - 1.0*x^9 + 1.0*x^10)
23:  Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + x^20 + x^21 + x^22)
24:  Poly(1.0 - 1.0*x^4 + 1.0*x^8)
25:  Poly(1.0 + 1.0*x^5 + 1.0*x^10 + 1.0*x^15 + 1.0*x^20)
26:  Poly(1.0 - 1.0*x + 1.0*x^2 - 1.0*x^3 + 1.0*x^4 - 1.0*x^5 + 1.0*x^6 - 1.0*x^7 + 1.0*x^8 - 1.0*x^9 + 1.0*x^10 - 1.0*x^11 + 1.0*x^12)
27:  Poly(1.0 + 1.0*x^9 + 1.0*x^18)
28:  Poly(1.0 - 1.0*x^2 + 1.0*x^4 - 1.0*x^6 + 1.0*x^8 - 1.0*x^10 + 1.0*x^12)
29:  Poly(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + x^20 + x^21 + x^22 + x^23 + x^24 + x^25 + x^26 + x^27 + x^28)
30:  Poly(1.0 + 1.0*x - 1.0*x^3 - 1.0*x^4 - 1.0*x^5 + 1.0*x^7 + 1.0*x^8)
The cyclotomic polynomial Φ(1) has a coefficient that is -1
The cyclotomic polynomial Φ(105) has a coefficient that is -2
The cyclotomic polynomial Φ(385) has a coefficient that is -3
The cyclotomic polynomial Φ(1365) has a coefficient that is -4
The cyclotomic polynomial Φ(1785) has a coefficient that is 5
The cyclotomic polynomial Φ(2805) has a coefficient that is -6
The cyclotomic polynomial Φ(3135) has a coefficient that is 7
The cyclotomic polynomial Φ(6545) has a coefficient that is -8
The cyclotomic polynomial Φ(6545) has a coefficient that is 9
The cyclotomic polynomial Φ(10465) has a coefficient that is 10
```

## Kotlin

Translation of: Java
`import java.util.TreeMapimport kotlin.math.absimport kotlin.math.powimport kotlin.math.sqrt private const val algorithm = 2 fun main() {    println("Task 1:  cyclotomic polynomials for n <= 30:")    for (i in 1..30) {        val p = cyclotomicPolynomial(i)        println("CP[\$i] = \$p")    }    println()     println("Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:")    var n = 0    for (i in 1..10) {        while (true) {            n++            val cyclo = cyclotomicPolynomial(n)            if (cyclo!!.hasCoefficientAbs(i)) {                println("CP[\$n] has coefficient with magnitude = \$i")                n--                break            }        }    }} private val COMPUTED: MutableMap<Int, Polynomial> = HashMap()private fun cyclotomicPolynomial(n: Int): Polynomial? {    if (COMPUTED.containsKey(n)) {        return COMPUTED[n]    }    if (n == 1) {        //  Polynomial:  x - 1        val p = Polynomial(1, 1, -1, 0)        COMPUTED[1] = p        return p    }    val factors = getFactors(n)    if (factors.containsKey(n)) {        //  n prime        val termList: MutableList<Term> = ArrayList()        for (index in 0 until n) {            termList.add(Term(1, index.toLong()))        }        val cyclo = Polynomial(termList)        COMPUTED[n] = cyclo        return cyclo    } else if (factors.size == 2 && factors.containsKey(2) && factors[2] == 1 && factors.containsKey(n / 2) && factors[n / 2] == 1) {        //  n = 2p        val prime = n / 2        val termList: MutableList<Term> = ArrayList()        var coeff = -1        for (index in 0 until prime) {            coeff *= -1            termList.add(Term(coeff.toLong(), index.toLong()))        }        val cyclo = Polynomial(termList)        COMPUTED[n] = cyclo        return cyclo    } else if (factors.size == 1 && factors.containsKey(2)) {        //  n = 2^h        val h = factors[2]!!        val termList: MutableList<Term> = ArrayList()        termList.add(Term(1, 2.0.pow((h - 1).toDouble()).toLong()))        termList.add(Term(1, 0))        val cyclo = Polynomial(termList)        COMPUTED[n] = cyclo        return cyclo    } else if (factors.size == 1 && !factors.containsKey(n)) {        // n = p^k        var p = 0        for (prime in factors.keys) {            p = prime        }        val k = factors[p]!!        val termList: MutableList<Term> = ArrayList()        for (index in 0 until p) {            termList.add(Term(1, (index * p.toDouble().pow(k - 1.toDouble()).toInt()).toLong()))        }        val cyclo = Polynomial(termList)        COMPUTED[n] = cyclo        return cyclo    } else if (factors.size == 2 && factors.containsKey(2)) {        //  n = 2^h * p^k        var p = 0        for (prime in factors.keys) {            if (prime != 2) {                p = prime            }        }        val termList: MutableList<Term> = ArrayList()        var coeff = -1        val twoExp = 2.0.pow((factors[2]!!) - 1.toDouble()).toInt()        val k = factors[p]!!        for (index in 0 until p) {            coeff *= -1            termList.add(Term(coeff.toLong(), (index * twoExp * p.toDouble().pow(k - 1.toDouble()).toInt()).toLong()))        }        val cyclo = Polynomial(termList)        COMPUTED[n] = cyclo        return cyclo    } else if (factors.containsKey(2) && n / 2 % 2 == 1 && n / 2 > 1) {        //  CP(2m)[x] = CP(-m)[x], n odd integer > 1        val cycloDiv2 = cyclotomicPolynomial(n / 2)        val termList: MutableList<Term> = ArrayList()        for (term in cycloDiv2!!.polynomialTerms) {            termList.add(if (term.exponent % 2 == 0L) term else term.negate())        }        val cyclo = Polynomial(termList)        COMPUTED[n] = cyclo        return cyclo    }     //  General Case    return when (algorithm) {        0 -> {            //  Slow - uses basic definition.            val divisors = getDivisors(n)            //  Polynomial:  ( x^n - 1 )            var cyclo = Polynomial(1, n, -1, 0)            for (i in divisors) {                val p = cyclotomicPolynomial(i)                cyclo = cyclo.divide(p)            }            COMPUTED[n] = cyclo            cyclo        }        1 -> {            //  Faster.  Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor            val divisors = getDivisors(n)            var maxDivisor = Int.MIN_VALUE            for (div in divisors) {                maxDivisor = maxDivisor.coerceAtLeast(div)            }            val divisorsExceptMax: MutableList<Int> = ArrayList()            for (div in divisors) {                if (maxDivisor % div != 0) {                    divisorsExceptMax.add(div)                }            }             //  Polynomial:  ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor            var cyclo = Polynomial(1, n, -1, 0).divide(Polynomial(1, maxDivisor, -1, 0))            for (i in divisorsExceptMax) {                val p = cyclotomicPolynomial(i)                cyclo = cyclo.divide(p)            }            COMPUTED[n] = cyclo            cyclo        }        2 -> {            //  Fastest            //  Let p ; q be primes such that p does not divide n, and q q divides n.            //  Then CP(np)[x] = CP(n)[x^p] / CP(n)[x]            var m = 1            var cyclo = cyclotomicPolynomial(m)            val primes = factors.keys.toMutableList()            primes.sort()            for (prime in primes) {                //  CP(m)[x]                val cycloM = cyclo                //  Compute CP(m)[x^p].                val termList: MutableList<Term> = ArrayList()                for (t in cycloM!!.polynomialTerms) {                    termList.add(Term(t.coefficient, t.exponent * prime))                }                cyclo = Polynomial(termList).divide(cycloM)                m *= prime            }            //  Now, m is the largest square free divisor of n            val s = n / m            //  Compute CP(n)[x] = CP(m)[x^s]            val termList: MutableList<Term> = ArrayList()            for (t in cyclo!!.polynomialTerms) {                termList.add(Term(t.coefficient, t.exponent * s))            }            cyclo = Polynomial(termList)            COMPUTED[n] = cyclo            cyclo        }        else -> {            throw RuntimeException("ERROR 103:  Invalid algorithm.")        }    }} private fun getDivisors(number: Int): List<Int> {    val divisors: MutableList<Int> = ArrayList()    val sqrt = sqrt(number.toDouble()).toLong()    for (i in 1..sqrt) {        if (number % i == 0L) {            divisors.add(i.toInt())            val div = (number / i).toInt()            if (div.toLong() != i && div != number) {                divisors.add(div)            }        }    }    return divisors} private fun crutch(): MutableMap<Int, Map<Int, Int>> {    val allFactors: MutableMap<Int, Map<Int, Int>> = TreeMap()     val factors: MutableMap<Int, Int> = TreeMap()    factors[2] = 1     allFactors[2] = factors    return allFactors} private val allFactors = crutch() var MAX_ALL_FACTORS = 100000 fun getFactors(number: Int): Map<Int, Int> {    if (allFactors.containsKey(number)) {        return allFactors[number]!!    }    val factors: MutableMap<Int, Int> = TreeMap()    if (number % 2 == 0) {        val factorsDivTwo = getFactors(number / 2)        factors.putAll(factorsDivTwo)        factors.merge(2, 1) { a: Int?, b: Int? -> Integer.sum(a!!, b!!) }        if (number < MAX_ALL_FACTORS) allFactors[number] = factors        return factors    }    val sqrt = sqrt(number.toDouble()).toLong()    var i = 3    while (i <= sqrt) {        if (number % i == 0) {            factors.putAll(getFactors(number / i))            factors.merge(i, 1) { a: Int?, b: Int? -> Integer.sum(a!!, b!!) }            if (number < MAX_ALL_FACTORS) {                allFactors[number] = factors            }            return factors        }        i += 2    }    factors[number] = 1    if (number < MAX_ALL_FACTORS) {        allFactors[number] = factors    }    return factors} private class Polynomial {    val polynomialTerms: MutableList<Term>     //  Format - coeff, exp, coeff, exp, (repeating in pairs) . . .    constructor(vararg values: Int) {        require(values.size % 2 == 0) { "ERROR 102:  Polynomial constructor.  Length must be even.  Length = " + values.size }        polynomialTerms = mutableListOf()        var i = 0        while (i < values.size) {            val t = Term(values[i].toLong(), values[i + 1].toLong())            polynomialTerms.add(t)            i += 2        }        polynomialTerms.sortWith(TermSorter())    }     constructor() {        //  zero        polynomialTerms = ArrayList()        polynomialTerms.add(Term(0, 0))    }     fun hasCoefficientAbs(coeff: Int): Boolean {        for (term in polynomialTerms) {            if (abs(term.coefficient) == coeff.toLong()) {                return true            }        }        return false    }     constructor(termList: MutableList<Term>) {        if (termList.isEmpty()) {            //  zero            termList.add(Term(0, 0))        } else {            //  Remove zero terms if needed            termList.removeIf { t -> t.coefficient == 0L }        }        if (termList.size == 0) {            //  zero            termList.add(Term(0, 0))        }        polynomialTerms = termList        polynomialTerms.sortWith(TermSorter())    }     fun divide(v: Polynomial?): Polynomial {        var q = Polynomial()        var r = this        val lcv = v!!.leadingCoefficient()        val dv = v.degree()        while (r.degree() >= v.degree()) {            val lcr = r.leadingCoefficient()            val s = lcr / lcv //  Integer division            val term = Term(s, r.degree() - dv)            q = q.add(term)            r = r.add(v.multiply(term.negate()))        }        return q    }     fun add(polynomial: Polynomial): Polynomial {        val termList: MutableList<Term> = ArrayList()        var thisCount = polynomialTerms.size        var polyCount = polynomial.polynomialTerms.size        while (thisCount > 0 || polyCount > 0) {            val thisTerm = if (thisCount == 0) null else polynomialTerms[thisCount - 1]            val polyTerm = if (polyCount == 0) null else polynomial.polynomialTerms[polyCount - 1]            when {                thisTerm == null -> {                    termList.add(polyTerm!!.clone())                    polyCount--                }                polyTerm == null -> {                    termList.add(thisTerm.clone())                    thisCount--                }                thisTerm.degree() == polyTerm.degree() -> {                    val t = thisTerm.add(polyTerm)                    if (t.coefficient != 0L) {                        termList.add(t)                    }                    thisCount--                    polyCount--                }                thisTerm.degree() < polyTerm.degree() -> {                    termList.add(thisTerm.clone())                    thisCount--                }                else -> {                    termList.add(polyTerm.clone())                    polyCount--                }            }        }        return Polynomial(termList)    }     fun add(term: Term): Polynomial {        val termList: MutableList<Term> = ArrayList()        var added = false        for (currentTerm in polynomialTerms) {            if (currentTerm.exponent == term.exponent) {                added = true                if (currentTerm.coefficient + term.coefficient != 0L) {                    termList.add(currentTerm.add(term))                }            } else {                termList.add(currentTerm.clone())            }        }        if (!added) {            termList.add(term.clone())        }        return Polynomial(termList)    }     fun multiply(term: Term): Polynomial {        val termList: MutableList<Term> = ArrayList()        for (currentTerm in polynomialTerms) {            termList.add(currentTerm.clone().multiply(term))        }        return Polynomial(termList)    }     fun leadingCoefficient(): Long {        return polynomialTerms[0].coefficient    }     fun degree(): Long {        return polynomialTerms[0].exponent    }     override fun toString(): String {        val sb = StringBuilder()        var first = true        for (term in polynomialTerms) {            if (first) {                sb.append(term)                first = false            } else {                sb.append(" ")                if (term.coefficient > 0) {                    sb.append("+ ")                    sb.append(term)                } else {                    sb.append("- ")                    sb.append(term.negate())                }            }        }        return sb.toString()    }} private class TermSorter : Comparator<Term> {    override fun compare(o1: Term, o2: Term): Int {        return (o2.exponent - o1.exponent).toInt()    }} //  Note:  Cyclotomic Polynomials have small coefficients.  Not appropriate for general polynomial usage.private class Term(var coefficient: Long, var exponent: Long) {    fun clone(): Term {        return Term(coefficient, exponent)    }     fun multiply(term: Term): Term {        return Term(coefficient * term.coefficient, exponent + term.exponent)    }     fun add(term: Term): Term {        if (exponent != term.exponent) {            throw RuntimeException("ERROR 102:  Exponents not equal.")        }        return Term(coefficient + term.coefficient, exponent)    }     fun negate(): Term {        return Term(-coefficient, exponent)    }     fun degree(): Long {        return exponent    }     override fun toString(): String {        if (coefficient == 0L) {            return "0"        }        if (exponent == 0L) {            return "" + coefficient        }        if (coefficient == 1L) {            return if (exponent == 1L) {                "x"            } else {                "x^\$exponent"            }        }        return if (exponent == 1L) {            coefficient.toString() + "x"        } else coefficient.toString() + "x^" + exponent    }}`
Output:
```Task 1:  cyclotomic polynomials for n <= 30:
CP[1] = x - 1
CP[2] = x + 1
CP[3] = x^2 + x + 1
CP[4] = x^2 + 1
CP[5] = x^4 + x^3 + x^2 + x + 1
CP[6] = x^2 - x + 1
CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[8] = x^4 + 1
CP[9] = x^6 + x^3 + 1
CP[10] = x^4 - x^3 + x^2 - x + 1
CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[12] = x^4 - x^2 + 1
CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
CP[16] = x^8 + 1
CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[18] = x^6 - x^3 + 1
CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[20] = x^8 - x^6 + x^4 - x^2 + 1
CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[24] = x^8 - x^4 + 1
CP[25] = x^20 + x^15 + x^10 + x^5 + 1
CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[27] = x^18 + x^9 + 1
CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:
CP[1] has coefficient with magnitude = 1
CP[105] has coefficient with magnitude = 2
CP[385] has coefficient with magnitude = 3
CP[1365] has coefficient with magnitude = 4
CP[1785] has coefficient with magnitude = 5
CP[2805] has coefficient with magnitude = 6
CP[3135] has coefficient with magnitude = 7
CP[6545] has coefficient with magnitude = 8
CP[6545] has coefficient with magnitude = 9
CP[10465] has coefficient with magnitude = 10```

## Maple

`with(NumberTheory):for n to 30 do lprint(Phi(n,x)) od: x-1x+1x^2+x+1x^2+1x^4+x^3+x^2+x+1x^2-x+1x^6+x^5+x^4+x^3+x^2+x+1x^4+1x^6+x^3+1x^4-x^3+x^2-x+1x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1x^4-x^2+1x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1x^6-x^5+x^4-x^3+x^2-x+1x^8-x^7+x^5-x^4+x^3-x+1x^8+1x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1x^6-x^3+1x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1x^8-x^6+x^4-x^2+1x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1x^8-x^4+1x^20+x^15+x^10+x^5+1x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1x^18+x^9+1x^12-x^10+x^8-x^6+x^4-x^2+1x^28+x^27+x^26+x^25+x^24+x^23+x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1x^8+x^7-x^5-x^4-x^3+x+1 PhiSet:=[seq(map(abs,{coeffs(Phi(k,x),x)}),k=1..15000)]:[seq(ListTools:-SelectFirst(s->member(n,s),PhiSet,output=indices),n=1..20)];#[1, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, #  10465, 10465, 10465, 11305, 11305, 11305, 11305, 11305, 11305]`

## Mathematica / Wolfram Language

`Cyclotomic[#, x] & /@ Range[30] // Columni = 1;n = 10;PrintTemporary[Dynamic[{magnitudes, i}]];magnitudes = ConstantArray[True, n];While[Or @@ magnitudes, coeff = Abs[CoefficientList[Cyclotomic[i, x], x]]; coeff = Select[coeff, Between[{1, n}]]; coeff = DeleteDuplicates[coeff]; If[Or @@ magnitudes[[coeff]],  Do[   If[magnitudes[[c]] == True,    Print["CyclotomicPolynomial(", i,      ") has coefficient with magnitude ", c]    ]   ,   {c, coeff}   ];  magnitudes[[coeff]] = False;  ]; i++; ]`
Output:
```-1+x
1+x
1+x+x^2
1+x^2
1+x+x^2+x^3+x^4
1-x+x^2
1+x+x^2+x^3+x^4+x^5+x^6
1+x^4
1+x^3+x^6
1-x+x^2-x^3+x^4
1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10
1-x^2+x^4
1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12
1-x+x^2-x^3+x^4-x^5+x^6
1-x+x^3-x^4+x^5-x^7+x^8
1+x^8
1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16
1-x^3+x^6
1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18
1-x^2+x^4-x^6+x^8
1-x+x^3-x^4+x^6-x^8+x^9-x^11+x^12
1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9+x^10
1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22
1-x^4+x^8
1+x^5+x^10+x^15+x^20
1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9+x^10-x^11+x^12
1+x^9+x^18
1-x^2+x^4-x^6+x^8-x^10+x^12
1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18+x^19+x^20+x^21+x^22+x^23+x^24+x^25+x^26+x^27+x^28
1+x-x^3-x^4-x^5+x^7+x^8

CyclotomicPolynomial(1) has coefficient with magnitude 1
CyclotomicPolynomial(105) has coefficient with magnitude 2
CyclotomicPolynomial(385) has coefficient with magnitude 3
CyclotomicPolynomial(1365) has coefficient with magnitude 4
CyclotomicPolynomial(1785) has coefficient with magnitude 5
CyclotomicPolynomial(2805) has coefficient with magnitude 6
CyclotomicPolynomial(3135) has coefficient with magnitude 7
CyclotomicPolynomial(6545) has coefficient with magnitude 8
CyclotomicPolynomial(6545) has coefficient with magnitude 9
CyclotomicPolynomial(10465) has coefficient with magnitude 10```

## Nim

Translation of: Java

We use Java algorithm with ideas from C#, D, Go and Kotlin. We have kept only algorithm number 2 as other algorithms are much less efficient. We have also done some Nim specific improvements in order to get better performances.

`import algorithm, math, sequtils, strformat, tables type   Term = tuple[coeff: int; exp: Natural]  Polynomial = seq[Term]   # Table used to represent the list of factors of a number.  # If, for a number "n", "k" is present in the table "f" of its factors,  # "f[k]" contains the exponent of "k" in the prime factor decomposition.  Factors = Table[int, int]  ##################################################################################################### Miscellaneous. ## Parity tests.template isOdd(n: int): bool = (n and 1) != 0template isEven(n: int): bool = (n and 1) == 0 #--------------------------------------------------------------------------------------------------- proc sort(poly: var Polynomial) {.inline.} =  ## Sort procedure for the terms of a polynomial (high degree first).  algorithm.sort(poly, proc(x, y: Term): int = cmp(x.exp, y.exp), Descending)  ##################################################################################################### Superscripts. const Superscripts: array['0'..'9', string] = ["⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"] func superscript(n: Natural): string =  ## Return the Unicode string to use to represent an exponent.  if n == 1:    return ""  for d in \$n:    result.add(Superscripts[d])  ##################################################################################################### Term operations. func term(coeff, exp: int): Term =  ## Create a term.  if exp < 0:    raise newException(ValueError, "term exponent cannot be negative")  (coeff, Natural exp) #--------------------------------------------------------------------------------------------------- func `*`(a, b: Term): Term =  ## Multiply two terms.  (a.coeff * b.coeff, Natural a.exp + b.exp) #--------------------------------------------------------------------------------------------------- func `+`(a, b: Term): Term =  ## Add two terms.   if a.exp != b.exp:    raise newException(ValueError, "addition of terms with unequal exponents")  (a.coeff + b.coeff, a.exp) #--------------------------------------------------------------------------------------------------- func `-`(a: Term): Term =  ## Return the opposite of a term.  (-a.coeff, a.exp) #--------------------------------------------------------------------------------------------------- func `\$`(a: Term): string =  ## Return the string representation of a term.  if a.coeff == 0: "0"  elif a.exp == 0: \$a.coeff  elif a.coeff == 1: 'x' & superscript(a.exp)  elif a.coeff == -1: "-x" & superscript(a.exp)  else: \$a.coeff & 'x' & superscript(a.exp)  ##################################################################################################### Polynomial. func polynomial(terms: varargs[Term]): Polynomial =  ## Create a polynomial described by its terms.  for t in terms:    if t.coeff != 0:      result.add(t)  if result.len == 0:    return @[term(0, 0)]  sort(result) #--------------------------------------------------------------------------------------------------- func hasCoeffAbs(poly: Polynomial; coeff: int): bool =  ## Return true if the polynomial contains a given coefficient.  for t in poly:    if abs(t.coeff) == coeff:      return true #--------------------------------------------------------------------------------------------------- func leadingCoeff(poly: Polynomial): int {.inline.} =  ## Return the coefficient of the term with the highest degree.  poly[0].coeff #--------------------------------------------------------------------------------------------------- func degree(poly: Polynomial): int {.inline.} =  ## Return the degree of the polynomial.  if poly.len == 0: -1  else: poly[0].exp #--------------------------------------------------------------------------------------------------- func `+`(poly: Polynomial; someTerm: Term): Polynomial =  ## Add a term to a polynomial.   var added = false  for currTerm in poly:    if currterm.exp == someTerm.exp:      added = true      if currTerm.coeff + someTerm.coeff != 0:        result.add(currTerm + someTerm)    else:      result.add(currTerm)   if not added:    result.add(someTerm) #--------------------------------------------------------------------------------------------------- func `+`(a, b: Polynomial): Polynomial =  ## Add two polynomials.   var aIndex = a.high  var bIndex = b.high   while aIndex >= 0 or bIndex >= 0:    if aIndex < 0:      result &= b[bIndex]      dec bIndex    elif bIndex < 0:      result &= a[aIndex]      dec aIndex    else:      let t1 = a[aIndex]      let t2 = b[bIndex]      if t1.exp == t2.exp:        let t3 = t1 + t2        if t3.coeff != 0:          result.add(t3)        dec aIndex        dec bIndex      elif t1.exp < t2.exp:        result.add(t1)        dec aIndex      else:        result.add(t2)        dec bIndex   sort(result) #--------------------------------------------------------------------------------------------------- func `*`(poly: Polynomial; someTerm: Term): Polynomial =  ## Multiply a polynomial by a term.  for currTerm in poly:    result.add(currTerm * someTerm) #--------------------------------------------------------------------------------------------------- func `/`(a, b: Polynomial): Polynomial =  ## Divide a polynomial by another polynomial.   var a = a  let lcb = b.leadingCoeff  let db = b.degree  while a.degree >= b.degree:    let lca = a.leadingCoeff    let s = lca div lcb    let t = term(s, a.degree - db)    result = result + t    a = a + b * -t #--------------------------------------------------------------------------------------------------- func `\$`(poly: Polynomial): string =  ## Return the string representation of a polynomial.   for t in poly:    if result.len == 0:      result.add(\$t)    else:      if t.coeff > 0:        result.add('+')        result.add(\$t)      else:        result.add('-')        result.add(\$(-t))  ##################################################################################################### Cyclotomic polynomial. var   # Cache of list of factors.  factorCache: Table[int, Factors] = {2: {2: 1}.toTable}.toTable   # Cache of cyclotomic polynomials. Initialized with 1 -> x - 1.  polyCache: Table[int, Polynomial] = {1: polynomial(term(1, 1), term(-1, 0))}.toTable #--------------------------------------------------------------------------------------------------- proc getFactors(n: int): Factors =  ## Return the list of factors of a number.   if n in factorCache:    return factorCache[n]   if n.isEven:    result = getFactors(n div 2)    result[2] = result.getOrDefault(2) + 1    factorCache[n] = result    return   var i = 3  while i * i <= n:    if n mod i == 0:      result = getFactors( n div i)      result[i] = result.getOrDefault(i) + 1      factorCache[n] = result      return    inc i, 2   result[n] = 1  factorCache[n] = result #--------------------------------------------------------------------------------------------------- proc cycloPoly(n: int): Polynomial =  ## Find the nth cyclotomic polynomial.   if n in polyCache:    return polyCache[n]   let factors = getFactors(n)   if n in factors:    # n is prime.    for i in countdown(n - 1, 0):       # Add the terms by decreasing degrees.      result.add(term(1, i))   elif factors.len == 2 and factors.getOrDefault(2) == 1 and factors.getOrDefault(n div 2) == 1:    # n = 2 x prime.    let prime = n div 2    var coeff = -1    for i in countdown(prime - 1, 0):   # Add the terms by decreasing degrees.      coeff *= -1      result.add(term(coeff, i))   elif factors.len == 1 and 2 in factors:    # n = 2 ^ h.    let h = factors[2]    result.add([term(1, 1 shl (h - 1)), term(1, 0)])   elif factors.len == 1 and n notin factors:    # n = prime ^ k.    var p, k = 0    for prime, v in factors.pairs:      if prime > p:        p = prime        k = v    for i in countdown(p - 1, 0):       # Add the terms by decreasing degrees.      result.add(term(1, i * p^(k-1)))   elif factors.len == 2 and 2 in factors:    # n = 2 ^ h x prime ^ k.    var p, k = 0    for prime, v in factors.pairs:      if prime != 2 and prime > p:        p = prime        k = v    var coeff = -1    let twoExp = 1 shl (factors[2] - 1)    for i in countdown(p - 1, 0):       # Add the terms by decreasing degrees.      coeff *= -1      result.add(term(coeff, i * twoExp * p^(k-1)))   elif 2 in factors and isOdd(n div 2) and n div 2 > 1:    # CP(2m)[x] = CP(-m)[x], n odd integer > 1.    let cycloDiv2 = cycloPoly(n div 2)    for t in cycloDiv2:      result.add(if t.exp.isEven: t else: -t)   else:    # Let p, q be primes such that p does not divide n, and q divides n.    # Then CP(np)[x] = CP(n)[x^p] / CP(n)[x].    var m = 1    var cyclo = cycloPoly(m)    let primes = sorted(toSeq(factors.keys))    for prime in primes:      # Compute CP(m)[x^p].      var terms: Polynomial      for t in cyclo:        terms.add(term(t.coeff, t.exp * prime))      cyclo = terms / cyclo      m *= prime    # Now, m is the largest square free divisor of n.    let s = n div m    # Compute CP(n)[x] = CP(m)[x^s].    for t in cyclo:      result.add(term(t.coeff, t.exp * s))   polyCache[n] = result  #——————————————————————————————————————————————————————————————————————————————————————————————————— echo "Cyclotomic polynomials for n ⩽ 30:"for i in 1..30:  echo &"Φ{'(' & \$i & ')':4} = {cycloPoly(i)}" echo ""echo "Smallest cyclotomic polynomial with n or -n as a coefficient:"var n = 0for i in 1..10:  while true:    inc n    if cycloPoly(n).hasCoeffAbs(i):      echo &"Φ{'(' & \$n & ')':7} has coefficient with magnitude = {i}"      dec n      break`
Output:

The program runs in 41 seconds on our reasonably performing laptop.

```Cyclotomic polynomials for n ⩽ 30:
Φ(1)  = x-1
Φ(2)  = x+1
Φ(3)  = x²+x+1
Φ(4)  = x²+1
Φ(5)  = x⁴+x³+x²+x+1
Φ(6)  = x²-x+1
Φ(7)  = x⁶+x⁵+x⁴+x³+x²+x+1
Φ(8)  = x⁴+1
Φ(9)  = x⁶+x³+1
Φ(10) = x⁴-x³+x²-x+1
Φ(11) = x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1
Φ(12) = x⁴-x²+1
Φ(13) = x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1
Φ(14) = x⁶-x⁵+x⁴-x³+x²-x+1
Φ(15) = x⁸-x⁷+x⁵-x⁴+x³-x+1
Φ(16) = x⁸+1
Φ(17) = x¹⁶+x¹⁵+x¹⁴+x¹³+x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1
Φ(18) = x⁶-x³+1
Φ(19) = x¹⁸+x¹⁷+x¹⁶+x¹⁵+x¹⁴+x¹³+x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1
Φ(20) = x⁸-x⁶+x⁴-x²+1
Φ(21) = x¹²-x¹¹+x⁹-x⁸+x⁶-x⁴+x³-x+1
Φ(22) = x¹⁰-x⁹+x⁸-x⁷+x⁶-x⁵+x⁴-x³+x²-x+1
Φ(23) = x²²+x²¹+x²⁰+x¹⁹+x¹⁸+x¹⁷+x¹⁶+x¹⁵+x¹⁴+x¹³+x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1
Φ(24) = x⁸-x⁴+1
Φ(25) = x²⁰+x¹⁵+x¹⁰+x⁵+1
Φ(26) = x¹²-x¹¹+x¹⁰-x⁹+x⁸-x⁷+x⁶-x⁵+x⁴-x³+x²-x+1
Φ(27) = x¹⁸+x⁹+1
Φ(28) = x¹²-x¹⁰+x⁸-x⁶+x⁴-x²+1
Φ(29) = x²⁸+x²⁷+x²⁶+x²⁵+x²⁴+x²³+x²²+x²¹+x²⁰+x¹⁹+x¹⁸+x¹⁷+x¹⁶+x¹⁵+x¹⁴+x¹³+x¹²+x¹¹+x¹⁰+x⁹+x⁸+x⁷+x⁶+x⁵+x⁴+x³+x²+x+1
Φ(30) = x⁸+x⁷-x⁵-x⁴-x³+x+1

Smallest cyclotomic polynomial with n or -n as a coefficient:
Φ(1)     has coefficient with magnitude = 1
Φ(105)   has coefficient with magnitude = 2
Φ(385)   has coefficient with magnitude = 3
Φ(1365)  has coefficient with magnitude = 4
Φ(1785)  has coefficient with magnitude = 5
Φ(2805)  has coefficient with magnitude = 6
Φ(3135)  has coefficient with magnitude = 7
Φ(6545)  has coefficient with magnitude = 8
Φ(6545)  has coefficient with magnitude = 9
Φ(10465) has coefficient with magnitude = 10```

## PARI/GP

Cyclotomic polynomials are a built-in function.

` for(n=1,30,print(n," : ",polcyclo(n))) contains_coeff(n, d) = p=polcyclo(n);for(k=0,poldegree(p),if(abs(polcoef(p,k))==d,return(1)));return(0) for(d=1,10,i=1; while(contains_coeff(i,d)==0,i=i+1);print(d," : ",i)) `
Output:
```
1 : x - 1
2 : x + 1
3 : x^2 + x + 1
4 : x^2 + 1
5 : x^4 + x^3 + x^2 + x + 1
6 : x^2 - x + 1
7 : x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
8 : x^4 + 1
9 : x^6 + x^3 + 1
10 : x^4 - x^3 + x^2 - x + 1
11 : x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
12 : x^4 - x^2 + 1
13 : x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
14 : x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
15 : x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
16 : x^8 + 1
17 : x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
18 : x^6 - x^3 + 1
19 : x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
20 : x^8 - x^6 + x^4 - x^2 + 1
21 : x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
22 : x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
23 : x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
24 : x^8 - x^4 + 1
25 : x^20 + x^15 + x^10 + x^5 + 1
26 : x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
27 : x^18 + x^9 + 1
28 : x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
29 : x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
30 : x^8 + x^7 - x^5 - x^4 - x^3 + x + 1
1 : 1
2 : 105
3 : 385
4 : 1365
5 : 1785
6 : 2805
7 : 3135
8 : 6545
9 : 6545
10 : 10465

```

## Perl

Conveniently, the module `Math::Polynomial::Cyclotomic` exists to do all the work. An `exponent too large` error prevents reaching the 10th step of the 2nd part of the task.

`use feature 'say';use List::Util qw(first);use Math::Polynomial::Cyclotomic qw(cyclo_poly_iterate); say 'First 30 cyclotomic polynomials:';my \$it = cyclo_poly_iterate(1);say "\$_: " . \$it->() for 1 .. 30; say "\nSmallest cyclotomic polynomial with n or -n as a coefficient:";\$it = cyclo_poly_iterate(1); for (my (\$n, \$k) = (1, 1) ; \$n <= 10 ; ++\$k) {    my \$poly = \$it->();    while (my \$c = first { abs(\$_) == \$n } \$poly->coeff) {        say "CP \$k has coefficient with magnitude = \$n";        \$n++;    }}`
Output:
```First 30 cyclotomic polynomials:
1: (x - 1)
2: (x + 1)
3: (x^2 + x + 1)
4: (x^2 + 1)
5: (x^4 + x^3 + x^2 + x + 1)
6: (x^2 - x + 1)
7: (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
8: (x^4 + 1)
9: (x^6 + x^3 + 1)
10: (x^4 - x^3 + x^2 - x + 1)
11: (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
12: (x^4 - x^2 + 1)
13: (x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
14: (x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
15: (x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)
16: (x^8 + 1)
17: (x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
18: (x^6 - x^3 + 1)
19: (x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
20: (x^8 - x^6 + x^4 - x^2 + 1)
21: (x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1)
22: (x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
23: (x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
24: (x^8 - x^4 + 1)
25: (x^20 + x^15 + x^10 + x^5 + 1)
26: (x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
27: (x^18 + x^9 + 1)
28: (x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1)
29: (x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
30: (x^8 + x^7 - x^5 - x^4 - x^3 + x + 1)

Smallest cyclotomic polynomial with n or -n as a coefficient:
CP 1 has coefficient with magnitude = 1
CP 105 has coefficient with magnitude = 2
CP 385 has coefficient with magnitude = 3
CP 1365 has coefficient with magnitude = 4
CP 1785 has coefficient with magnitude = 5
CP 2805 has coefficient with magnitude = 6
CP 3135 has coefficient with magnitude = 7
CP 6545 has coefficient with magnitude = 8
CP 6545 has coefficient with magnitude = 9```

## Phix

Translation of: Julia

Uses several routines from Polynomial_long_division#Phix, tweaked slightly to check remainder is zero and trim the quotient.

```-- demo\rosetta\Cyclotomic_Polynomial.exw
with javascript_semantics
function degree(sequence p)
for i=length(p) to 1 by -1 do
if p[i]!=0 then return i end if
end for
return -1
end function

function poly_div(sequence n, d)
while length(d)<length(n) do d &=0 end while
integer dn = degree(n),
dd = degree(d)
if dd<0 then throw("divide by zero") end if
sequence quot = repeat(0,dn)
while dn>=dd do
integer k = dn-dd
integer qk = n[dn]/d[dd]
quot[k+1] = qk
sequence d2 = d[1..length(d)-k]
for i=1 to length(d2) do
integer mi = -i
n[mi] -= d2[mi]*qk
end for
dn = degree(n)
end while
--  return {quot,n} -- (n is now the remainder)
if n!=repeat(0,length(n)) then ?9/0 end if
while quot[\$]=0 do quot = quot[1..\$-1] end while
return quot
end function

function poly(sequence si)
-- display helper
string r = ""
for t=length(si) to 1 by -1 do
integer sit = si[t]
if sit!=0 then
if sit=1 and t>1 then
r &= iff(r=""? "":" + ")
elsif sit=-1 and t>1 then
r &= iff(r=""?"-":" - ")
else
if r!="" then
r &= iff(sit<0?" - ":" + ")
sit = abs(sit)
end if
r &= sprintf("%d",sit)
end if
r &= iff(t>1?"x"&iff(t>2?sprintf("^%d",t-1):""):"")
end if
end for
if r="" then r="0" end if
return r
end function
--</Polynomial_long_division.exw>

--# memoize cache for recursive calls
constant cyclotomics = new_dict({{1,{-1,1}},{2,{1,1}}})

function cyclotomic(integer n)
--
-- Calculate the nth cyclotomic polynomial.
-- See wikipedia article at bottom of section /wiki/Cyclotomic_polynomial#Fundamental_tools
-- The algorithm is reliable but slow for large n > 1000.
--
sequence c
if getd_index(n,cyclotomics)!=NULL then
c = getd(n,cyclotomics)
else
if is_prime(n) then
c = repeat(1,n)
else  -- recursive formula seen in wikipedia article
c = -1&repeat(0,n-1)&1
sequence f = factors(n,-1)
for i=1 to length(f) do
c = poly_div(c,deep_copy(cyclotomic(f[i])))
end for
end if
setd(n,c,cyclotomics)
end if
return c
end function

for i=1 to 30 do
sequence z = cyclotomic(i)
string s = poly(z)
printf(1,"cp(%2d) = %s\n",{i,s})
if i>1 and z!=reverse(z) then ?9/0 end if -- sanity check
end for

integer found = 0, n = 1, cheat = 0
sequence fn = repeat(false,10),
nxt = {105,385,1365,1785,2805,3135,6545,6545,10465,10465}
atom t1 = time()+1
puts(1,"\n")
while found<iff(platform()=JS?5:10) do
sequence z = cyclotomic(n)
for i=1 to length(z) do
atom azi = abs(z[i])
if azi>=1 and azi<=10 and fn[azi]=0 then
printf(1,"cp(%d) has a coefficient with magnitude %d\n",{n,azi})
cheat = azi -- (comment this out to prevent cheating!)
found += 1
fn[azi] = true
t1 = time()+1
end if
end for
if cheat then {n,cheat} = {nxt[cheat],0} else n += iff(n=1?4:10) end if
if time()>t1 and platform()!=JS then
printf(1,"working (%d) ...\r",n)
t1 = time()+1
end if
end while
```
Output:

If you disable the cheating, and if in a particularly self harming mood replace it with n+=1, you will get exactly the same output, eventually.
(The distributed version contains simple instrumentation showing cp(1260) executes the line in the heart of poly_div() that subtracts a multiple of qk over 15 million times.)

```cp( 1) = x - 1
cp( 2) = x + 1
cp( 3) = x^2 + x + 1
cp( 4) = x^2 + 1
cp( 5) = x^4 + x^3 + x^2 + x + 1
cp( 6) = x^2 - x + 1
cp( 7) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
cp( 8) = x^4 + 1
cp( 9) = x^6 + x^3 + 1
cp(10) = x^4 - x^3 + x^2 - x + 1
cp(11) = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
cp(12) = x^4 - x^2 + 1
cp(13) = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
cp(14) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
cp(15) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
cp(16) = x^8 + 1
cp(17) = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
cp(18) = x^6 - x^3 + 1
cp(19) = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
cp(20) = x^8 - x^6 + x^4 - x^2 + 1
cp(21) = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
cp(22) = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
cp(23) = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
cp(24) = x^8 - x^4 + 1
cp(25) = x^20 + x^15 + x^10 + x^5 + 1
cp(26) = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
cp(27) = x^18 + x^9 + 1
cp(28) = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
cp(29) = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
cp(30) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

cp(1) has a coefficient with magnitude 1
cp(105) has a coefficient with magnitude 2
cp(385) has a coefficient with magnitude 3
cp(1365) has a coefficient with magnitude 4
cp(1785) has a coefficient with magnitude 5
cp(2805) has a coefficient with magnitude 6
cp(3135) has a coefficient with magnitude 7
cp(6545) has a coefficient with magnitude 8
cp(6545) has a coefficient with magnitude 9
cp(10465) has a coefficient with magnitude 10
```

## Python

`from itertools import count, chainfrom collections import deque def primes(_cache=[2, 3]):    yield from _cache    for n in count(_cache[-1]+2, 2):        if isprime(n):            _cache.append(n)            yield n def isprime(n):    for p in primes():        if n%p == 0:            return False        if p*p > n:            return True def factors(n):    for p in primes():    # prime factoring is such a non-issue for small numbers that, for    # this example, we might even just say    # for p in count(2):        if p*p > n:            if n > 1:                yield(n, 1, 1)            break         if n%p == 0:            cnt = 0            while True:                n, cnt = n//p, cnt+1                if n%p != 0: break            yield p, cnt, n# ^^ not the most sophisticated prime number routines, because no need # Returns (list1, list2) representing the division between# two polinomials. A list p of integers means the product#   (x^p[0] - 1) * (x^p[1] - 1) * ...def cyclotomic(n):    def poly_div(num, den):        return (num[0] + den[1], num[1] + den[0])     def elevate(poly, n): # replace poly p(x) with p(x**n)        powerup = lambda p, n: [a*n for a in p]        return poly if n == 1 else (powerup(poly[0], n), powerup(poly[1], n))      if n == 0:        return ([], [])    if n == 1:        return ([1], [])     p, m, r = next(factors(n))    poly = cyclotomic(r)    return elevate(poly_div(elevate(poly, p), poly), p**(m-1)) def to_text(poly):    def getx(c, e):        if e == 0:            return '1'        elif e == 1:            return 'x'        return 'x' + (''.join('⁰¹²³⁴⁵⁶⁷⁸⁹'[i] for i in map(int, str(e))))     parts = []    for (c,e) in (poly):        if c < 0:            coef = ' - ' if c == -1 else f' - {-c} '        else:            coef = (parts and ' + ' or '') if c == 1 else f' + {c}'        parts.append(coef + getx(c,e))    return ''.join(parts) def terms(poly):    # convert above representation of division to (coef, power) pairs     def merge(a, b):        # a, b should be deques. They may change during the course.        while a or b:            l = a[0] if a else (0, -1) # sentinel value            r = b[0] if b else (0, -1)            if l[1] > r[1]:                a.popleft()            elif l[1] < r[1]:                b.popleft()                l = r            else:                a.popleft()                b.popleft()                l = (l[0] + r[0], l[1])            yield l     def mul(poly, p): # p means polynomial x^p - 1        poly = list(poly)        return merge(deque((c, e+p) for c,e in poly),                     deque((-c, e) for c,e in poly))     def div(poly, p): # p means polynomial x^p - 1        q = deque()        for c,e in merge(deque(poly), q):            if c:                q.append((c, e - p))                yield (c, e - p)            if e == p: break     p = [(1, 0)]  # 1*x^0, i.e. 1     for x in poly[0]: # numerator        p = mul(p, x)    for x in sorted(poly[1], reverse=True): # denominator        p = div(p, x)    return p for n in chain(range(11), [2]):    print(f'{n}: {to_text(terms(cyclotomic(n)))}') want = 1for n in count():    c = [c for c,_ in terms(cyclotomic(n))]    while want in c or -want in c:        print(f'C[{want}]: {n}')        want += 1`
Output:

Only showing first 10 polynomials to avoid clutter.

```0: 1
1: x - 1
2: x + 1
3: x² + x + 1
4: x² + 1
5: x⁴ + x³ + x² + x + 1
6: x² - x + 1
7: x⁶ + x⁵ + x⁴ + x³ + x² + x + 1
8: x⁴ + 1
9: x⁶ + x³ + 1
10: x⁴ - x³ + x² - x + 1
105: x⁴⁸ + x⁴⁷ + x⁴⁶ - x⁴³ - x⁴² - 2 x⁴¹ - x⁴⁰ - x³⁹ + x³⁶ + x³⁵ + x³⁴ + x³³ + x³² + x³¹ - x²⁸ - x²⁶ - x²⁴ - x²² - x²⁰ + x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² - x⁹ - x⁸ - 2 x⁷ - x⁶ - x⁵ + x² + x + 1
C[1]: 0
C[2]: 105
C[3]: 385
C[4]: 1365
C[5]: 1785
C[6]: 2805
C[7]: 3135
C[8]: 6545
C[9]: 6545
C[10]: 10465
C[11]: 10465
C[12]: 10465
C[13]: 10465
C[14]: 10465
C[15]: 11305
C[16]: 11305
C[17]: 11305
C[18]: 11305
C[19]: 11305
C[20]: 11305
C[21]: 11305
C[22]: 15015
C[23]: 15015
```

## Raku

(formerly Perl 6)

Works with: Rakudo version 2020.01
Translation of: Perl

Uses the same library as Perl, so comes with the same caveats.

`use Math::Polynomial::Cyclotomic:from<Perl5> <cyclo_poly_iterate cyclo_poly>; say 'First 30 cyclotomic polynomials:';my \$iterator = cyclo_poly_iterate(1);say "Φ(\$_) = " ~ super \$iterator().Str for 1..30; say "\nSmallest cyclotomic polynomial with |n| as a coefficient:";say "Φ(1) has a coefficient magnitude: 1"; my \$index = 0;for 2..9 -> \$coefficient {    loop {        \$index += 5;        my \Φ = cyclo_poly(\$index);        next unless Φ ~~ / \$coefficient\* /;        say "Φ(\$index) has a coefficient magnitude: \$coefficient";        \$index -= 5;        last;    }} sub super (\$str) {    \$str.subst( / '^' (\d+) /, { \$0.trans([<0123456789>.comb] => [<⁰¹²³⁴⁵⁶⁷⁸⁹>.comb]) }, :g)}`
```First 30 cyclotomic polynomials:
Φ(1) = (x - 1)
Φ(2) = (x + 1)
Φ(3) = (x² + x + 1)
Φ(4) = (x² + 1)
Φ(5) = (x⁴ + x³ + x² + x + 1)
Φ(6) = (x² - x + 1)
Φ(7) = (x⁶ + x⁵ + x⁴ + x³ + x² + x + 1)
Φ(8) = (x⁴ + 1)
Φ(9) = (x⁶ + x³ + 1)
Φ(10) = (x⁴ - x³ + x² - x + 1)
Φ(11) = (x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1)
Φ(12) = (x⁴ - x² + 1)
Φ(13) = (x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1)
Φ(14) = (x⁶ - x⁵ + x⁴ - x³ + x² - x + 1)
Φ(15) = (x⁸ - x⁷ + x⁵ - x⁴ + x³ - x + 1)
Φ(16) = (x⁸ + 1)
Φ(17) = (x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1)
Φ(18) = (x⁶ - x³ + 1)
Φ(19) = (x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1)
Φ(20) = (x⁸ - x⁶ + x⁴ - x² + 1)
Φ(21) = (x¹² - x¹¹ + x⁹ - x⁸ + x⁶ - x⁴ + x³ - x + 1)
Φ(22) = (x¹⁰ - x⁹ + x⁸ - x⁷ + x⁶ - x⁵ + x⁴ - x³ + x² - x + 1)
Φ(23) = (x²² + x²¹ + x²⁰ + x¹⁹ + x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1)
Φ(24) = (x⁸ - x⁴ + 1)
Φ(25) = (x²⁰ + x¹⁵ + x¹⁰ + x⁵ + 1)
Φ(26) = (x¹² - x¹¹ + x¹⁰ - x⁹ + x⁸ - x⁷ + x⁶ - x⁵ + x⁴ - x³ + x² - x + 1)
Φ(27) = (x¹⁸ + x⁹ + 1)
Φ(28) = (x¹² - x¹⁰ + x⁸ - x⁶ + x⁴ - x² + 1)
Φ(29) = (x²⁸ + x²⁷ + x²⁶ + x²⁵ + x²⁴ + x²³ + x²² + x²¹ + x²⁰ + x¹⁹ + x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x + 1)
Φ(30) = (x⁸ + x⁷ - x⁵ - x⁴ - x³ + x + 1)

Smallest cyclotomic polynomial with |n| as a coefficient:
Φ(1) has a coefficient magnitude: 1
Φ(105) has a coefficient magnitude: 2
Φ(385) has a coefficient magnitude: 3
Φ(1365) has a coefficient magnitude: 4
Φ(1785) has a coefficient magnitude: 5
Φ(2805) has a coefficient magnitude: 6
Φ(3135) has a coefficient magnitude: 7
Φ(6545) has a coefficient magnitude: 8
Φ(6545) has a coefficient magnitude: 9```

## Sidef

Solution based on polynomial interpolation (slow).

`var Poly = require('Math::Polynomial')Poly.string_config(Hash(fold_sign => true, prefix => "", suffix => "")) func poly_interpolation(v) {    v.len.of {|n| v.len.of {|k| n**k } }.msolve(v)} say "First 30 cyclotomic polynomials:"for k in (1..30) {    var a = (k+1).of { cyclotomic(k, _) }    var Φ = poly_interpolation(a)    say ("Φ(#{k}) = ", Poly.new(Φ...))} say "\nSmallest cyclotomic polynomial with n or -n as a coefficient:"for n in (1..10) {  # very slow    var k = (1..Inf -> first {|k|        poly_interpolation((k+1).of { cyclotomic(k, _) }).first { .abs == n }    })    say "Φ(#{k}) has coefficient with magnitude #{n}"}`

Slightly faster solution, using the Math::Polynomial::Cyclotomic Perl module.

`var Poly = require('Math::Polynomial')           require('Math::Polynomial::Cyclotomic') Poly.string_config(Hash(fold_sign => true, prefix => "", suffix => "")) say "First 30 cyclotomic polynomials:"for k in (1..30) {    say ("Φ(#{k}) = ", Poly.new.cyclotomic(k))} say "\nSmallest cyclotomic polynomial with n or -n as a coefficient:"for n in (1..10) {    var p = Poly.new    var k = (1..Inf -> first {|k|        [p.cyclotomic(k).coeff].first { .abs == n }    })    say "Φ(#{k}) has coefficient with magnitude = #{n}"}`
Output:
```First 30 cyclotomic polynomials:
Φ(1) = x - 1
Φ(2) = x + 1
Φ(3) = x^2 + x + 1
Φ(4) = x^2 + 1
Φ(5) = x^4 + x^3 + x^2 + x + 1
Φ(6) = x^2 - x + 1
Φ(7) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
Φ(8) = x^4 + 1
Φ(9) = x^6 + x^3 + 1
Φ(10) = x^4 - x^3 + x^2 - x + 1
Φ(11) = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
Φ(12) = x^4 - x^2 + 1
Φ(13) = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
Φ(14) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Φ(15) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Φ(16) = x^8 + 1
Φ(17) = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
Φ(18) = x^6 - x^3 + 1
Φ(19) = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
Φ(20) = x^8 - x^6 + x^4 - x^2 + 1
Φ(21) = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Φ(22) = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Φ(23) = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
Φ(24) = x^8 - x^4 + 1
Φ(25) = x^20 + x^15 + x^10 + x^5 + 1
Φ(26) = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Φ(27) = x^18 + x^9 + 1
Φ(28) = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Φ(29) = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
Φ(30) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Smallest cyclotomic polynomial with n or -n as a coefficient:
Φ(1) has coefficient with magnitude = 1
Φ(105) has coefficient with magnitude = 2
Φ(385) has coefficient with magnitude = 3
Φ(1365) has coefficient with magnitude = 4
Φ(1785) has coefficient with magnitude = 5
Φ(2805) has coefficient with magnitude = 6
Φ(3135) has coefficient with magnitude = 7
^C
```

## Visual Basic .NET

Translation of: C++
`Imports System.Text Module Module1    Private ReadOnly MAX_ALL_FACTORS As Integer = 100_000#Const ALGORITHM = 2     Class Term        Implements IComparable(Of Term)         Public ReadOnly Property Coefficient As Long        Public ReadOnly Property Exponent As Long         Public Sub New(c As Long, Optional e As Long = 0)            Coefficient = c            Exponent = e        End Sub         Public Shared Operator -(t As Term) As Term            Return New Term(-t.Coefficient, t.Exponent)        End Operator         Public Shared Operator +(lhs As Term, rhs As Term) As Term            If lhs.Exponent <> rhs.Exponent Then                Throw New ArgumentException("Exponents not equal")            End If            Return New Term(lhs.Coefficient + rhs.Coefficient, lhs.Exponent)        End Operator         Public Shared Operator *(lhs As Term, rhs As Term) As Term            Return New Term(lhs.Coefficient * rhs.Coefficient, lhs.Exponent + rhs.Exponent)        End Operator         Public Function CompareTo(other As Term) As Integer Implements IComparable(Of Term).CompareTo            Return -Exponent.CompareTo(other.Exponent)        End Function         Public Overrides Function ToString() As String            If Coefficient = 0 Then                Return "0"            End If            If Exponent = 0 Then                Return Coefficient.ToString            End If            If Coefficient = 1 Then                If Exponent = 1 Then                    Return "x"                End If                Return String.Format("x^{0}", Exponent)            End If            If Coefficient = -1 Then                If Exponent = 1 Then                    Return "-x"                End If                Return String.Format("-x^{0}", Exponent)            End If            If Exponent = 1 Then                Return String.Format("{0}x", Coefficient)            End If            Return String.Format("{0}x^{1}", Coefficient, Exponent)        End Function    End Class     Class Polynomial        Implements IEnumerable(Of Term)         Private ReadOnly polyTerms As New List(Of Term)         Public Sub New()            polyTerms.Add(New Term(0))        End Sub         Public Sub New(ParamArray values() As Term)            If values.Length = 0 Then                polyTerms.Add(New Term(0))            Else                polyTerms.AddRange(values)            End If            Normalize()        End Sub         Public Sub New(values As IEnumerable(Of Term))            polyTerms.AddRange(values)            If polyTerms.Count = 0 Then                polyTerms.Add(New Term(0))            End If            Normalize()        End Sub         Public Function LeadingCoeficient() As Long            Return polyTerms(0).Coefficient        End Function         Public Function Degree() As Long            Return polyTerms(0).Exponent        End Function         Public Function HasCoefficentAbs(coeff As Long) As Boolean            For Each t In polyTerms                If Math.Abs(t.Coefficient) = coeff Then                    Return True                End If            Next            Return False        End Function         Public Function GetEnumerator() As IEnumerator(Of Term) Implements IEnumerable(Of Term).GetEnumerator            Return polyTerms.GetEnumerator        End Function         Private Function IEnumerable_GetEnumerator() As IEnumerator Implements IEnumerable.GetEnumerator            Return polyTerms.GetEnumerator        End Function         Private Sub Normalize()            polyTerms.Sort(Function(a As Term, b As Term) a.CompareTo(b))        End Sub         Public Shared Operator +(lhs As Polynomial, rhs As Term) As Polynomial            Dim terms As New List(Of Term)            Dim added = False            For Each ct In lhs                If ct.Exponent = rhs.Exponent Then                    added = True                    If ct.Coefficient + rhs.Coefficient <> 0 Then                        terms.Add(ct + rhs)                    End If                Else                    terms.Add(ct)                End If            Next            If Not added Then                terms.Add(rhs)            End If            Return New Polynomial(terms)        End Operator         Public Shared Operator *(lhs As Polynomial, rhs As Term) As Polynomial            Dim terms As New List(Of Term)            For Each ct In lhs                terms.Add(ct * rhs)            Next            Return New Polynomial(terms)        End Operator         Public Shared Operator +(lhs As Polynomial, rhs As Polynomial) As Polynomial            Dim terms As New List(Of Term)            Dim thisCount = lhs.polyTerms.Count            Dim polyCount = rhs.polyTerms.Count            While thisCount > 0 OrElse polyCount > 0                If thisCount = 0 Then                    Dim polyTerm = rhs.polyTerms(polyCount - 1)                    terms.Add(polyTerm)                    polyCount -= 1                ElseIf polyCount = 0 Then                    Dim thisTerm = lhs.polyTerms(thisCount - 1)                    terms.Add(thisTerm)                    thisCount -= 1                Else                    Dim polyTerm = rhs.polyTerms(polyCount - 1)                    Dim thisTerm = lhs.polyTerms(thisCount - 1)                    If thisTerm.Exponent = polyTerm.Exponent Then                        Dim t = thisTerm + polyTerm                        If t.Coefficient <> 0 Then                            terms.Add(t)                        End If                        thisCount -= 1                        polyCount -= 1                    ElseIf thisTerm.Exponent < polyTerm.Exponent Then                        terms.Add(thisTerm)                        thisCount -= 1                    Else                        terms.Add(polyTerm)                        polyCount -= 1                    End If                End If            End While            Return New Polynomial(terms)        End Operator         Public Shared Operator *(lhs As Polynomial, rhs As Polynomial) As Polynomial            Throw New Exception("Not implemented")        End Operator         Public Shared Operator /(lhs As Polynomial, rhs As Polynomial) As Polynomial            Dim q As New Polynomial            Dim r = lhs            Dim lcv = rhs.LeadingCoeficient            Dim dv = rhs.Degree            While r.Degree >= rhs.Degree                Dim lcr = r.LeadingCoeficient                Dim s = lcr \ lcv                Dim t As New Term(s, r.Degree() - dv)                q += t                r += rhs * -t            End While            Return q        End Operator         Public Overrides Function ToString() As String            Dim builder As New StringBuilder            Dim it = polyTerms.GetEnumerator()            If it.MoveNext Then                builder.Append(it.Current)            End If            While it.MoveNext                If it.Current.Coefficient < 0 Then                    builder.Append(" - ")                    builder.Append(-it.Current)                Else                    builder.Append(" + ")                    builder.Append(it.Current)                End If            End While            Return builder.ToString        End Function    End Class     Function GetDivisors(number As Integer) As List(Of Integer)        Dim divisors As New List(Of Integer)        Dim root = CType(Math.Sqrt(number), Long)        For i = 1 To root            If number Mod i = 0 Then                divisors.Add(i)                Dim div = number \ i                If div <> i AndAlso div <> number Then                    divisors.Add(div)                End If            End If        Next        Return divisors    End Function     Private ReadOnly allFactors As New Dictionary(Of Integer, Dictionary(Of Integer, Integer)) From {{2, New Dictionary(Of Integer, Integer) From {{2, 1}}}}    Function GetFactors(number As Integer) As Dictionary(Of Integer, Integer)        If allFactors.ContainsKey(number) Then            Return allFactors(number)        End If         Dim factors As New Dictionary(Of Integer, Integer)        If number Mod 2 = 0 Then            Dim factorsDivTwo = GetFactors(number \ 2)            For Each pair In factorsDivTwo                If Not factors.ContainsKey(pair.Key) Then                    factors.Add(pair.Key, pair.Value)                End If            Next            If factors.ContainsKey(2) Then                factors(2) += 1            Else                factors.Add(2, 1)            End If            If number < MAX_ALL_FACTORS Then                allFactors.Add(number, factors)            End If            Return factors        End If        Dim root = CType(Math.Sqrt(number), Long)        Dim i = 3L        While i <= root            If number Mod i = 0 Then                Dim factorsDivI = GetFactors(number \ i)                For Each pair In factorsDivI                    If Not factors.ContainsKey(pair.Key) Then                        factors.Add(pair.Key, pair.Value)                    End If                Next                If factors.ContainsKey(i) Then                    factors(i) += 1                Else                    factors.Add(i, 1)                End If                If number < MAX_ALL_FACTORS Then                    allFactors.Add(number, factors)                End If                Return factors            End If            i += 2        End While        factors.Add(number, 1)        If number < MAX_ALL_FACTORS Then            allFactors.Add(number, factors)        End If        Return factors    End Function     Private ReadOnly computedPolynomials As New Dictionary(Of Integer, Polynomial)    Function CyclotomicPolynomial(n As Integer) As Polynomial        If computedPolynomials.ContainsKey(n) Then            Return computedPolynomials(n)        End If         If n = 1 Then            REM polynomial: x - 1            Dim p As New Polynomial(New Term(1, 1), New Term(-1))            computedPolynomials.Add(n, p)            Return p        End If         Dim factors = GetFactors(n)        Dim terms As New List(Of Term)        Dim cyclo As Polynomial         If factors.ContainsKey(n) Then            REM n prime            For index = 1 To n                terms.Add(New Term(1, index - 1))            Next             cyclo = New Polynomial(terms)            computedPolynomials.Add(n, cyclo)            Return cyclo        ElseIf factors.Count = 2 AndAlso factors.ContainsKey(2) AndAlso factors(2) = 1 AndAlso factors.ContainsKey(n / 2) AndAlso factors(n / 2) = 1 Then            REM n = 2p            Dim prime = n \ 2            Dim coeff = -1             For index = 1 To prime                coeff *= -1                terms.Add(New Term(coeff, index - 1))            Next             cyclo = New Polynomial(terms)            computedPolynomials.Add(n, cyclo)            Return cyclo        ElseIf factors.Count = 1 AndAlso factors.ContainsKey(2) Then            REM n = 2^h            Dim h = factors(2)            terms = New List(Of Term) From {                New Term(1, Math.Pow(2, h - 1)),                New Term(1)            }             cyclo = New Polynomial(terms)            computedPolynomials.Add(n, cyclo)            Return cyclo        ElseIf factors.Count = 1 AndAlso factors.ContainsKey(n) Then            REM n = p^k            Dim p = 0            Dim k = 0            For Each it In factors                p = it.Key                k = it.Value            Next            For index = 1 To p                terms.Add(New Term(1, (index - 1) * Math.Pow(p, k - 1)))            Next             cyclo = New Polynomial(terms)            computedPolynomials.Add(n, cyclo)            Return cyclo        ElseIf factors.Count = 2 AndAlso factors.ContainsKey(2) Then            REM n = 2^h * p^k            Dim p = 0            For Each it In factors                If it.Key <> 2 Then                    p = it.Key                End If            Next             Dim coeff = -1            Dim twoExp = CType(Math.Pow(2, factors(2) - 1), Long)            Dim k = factors(p)            For index = 1 To p                coeff *= -1                terms.Add(New Term(coeff, (index - 1) * twoExp * Math.Pow(p, k - 1)))            Next             cyclo = New Polynomial(terms)            computedPolynomials.Add(n, cyclo)            Return cyclo        ElseIf factors.ContainsKey(2) AndAlso (n / 2) Mod 2 = 1 AndAlso n / 2 > 1 Then            REM CP(2m)[x] = CP(-m)[x], n odd integer > 1            Dim cycloDiv2 = CyclotomicPolynomial(n \ 2)            For Each t In cycloDiv2                If t.Exponent Mod 2 = 0 Then                    terms.Add(t)                Else                    terms.Add(-t)                End If            Next             cyclo = New Polynomial(terms)            computedPolynomials.Add(n, cyclo)            Return cyclo        End If #If ALGORITHM = 0 Then        REM slow - uses basic definition        Dim divisors = GetDivisors(n)        REM Polynomial: (x^n - 1)        cyclo = New Polynomial(New Term(1, n), New Term(-1))        For Each i In divisors            Dim p = CyclotomicPolynomial(i)            cyclo /= p        Next         computedPolynomials.Add(n, cyclo)        Return cyclo#ElseIf ALGORITHM = 1 Then        REM Faster.  Remove Max divisor (and all divisors of max divisor) - only one divide for all divisors of Max Divisor        Dim divisors = GetDivisors(n)        Dim maxDivisor = Integer.MinValue        For Each div In divisors            maxDivisor = Math.Max(maxDivisor, div)        Next        Dim divisorExceptMax As New List(Of Integer)        For Each div In divisors            If maxDivisor Mod div <> 0 Then                divisorExceptMax.Add(div)            End If        Next         REM Polynomial:  ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor        cyclo = New Polynomial(New Term(1, n), New Term(-1)) / New Polynomial(New Term(1, maxDivisor), New Term(-1))        For Each i In divisorExceptMax            Dim p = CyclotomicPolynomial(i)            cyclo /= p        Next         computedPolynomials.Add(n, cyclo)        Return cyclo#ElseIf ALGORITHM = 2 Then        REM Fastest        REM Let p ; q be primes such that p does not divide n, and q divides n        REM Then Cp(np)[x] = CP(n)[x^p] / CP(n)[x]        Dim m = 1        cyclo = CyclotomicPolynomial(m)        Dim primes As New List(Of Integer)        For Each it In factors            primes.Add(it.Key)        Next        primes.Sort()        For Each prime In primes            REM CP(m)[x]            Dim cycloM = cyclo            REM Compute CP(m)[x^p]            terms = New List(Of Term)            For Each t In cyclo                terms.Add(New Term(t.Coefficient, t.Exponent * prime))            Next            cyclo = New Polynomial(terms) / cycloM            m *= prime        Next        REM Now, m is the largest square free divisor of n        Dim s = n \ m        REM Compute CP(n)[x] = CP(m)[x^s]        terms = New List(Of Term)        For Each t In cyclo            terms.Add(New Term(t.Coefficient, t.Exponent * s))        Next         cyclo = New Polynomial(terms)        computedPolynomials.Add(n, cyclo)        Return cyclo#Else        Throw New Exception("Invalid algorithm")#End If    End Function     Sub Main()        Console.WriteLine("Task 1:  cyclotomic polynomials for n <= 30:")        For i = 1 To 30            Dim p = CyclotomicPolynomial(i)            Console.WriteLine("CP[{0}] = {1}", i, p)        Next        Console.WriteLine()         Console.WriteLine("Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:")        Dim n = 0        For i = 1 To 10            While True                n += 1                Dim cyclo = CyclotomicPolynomial(n)                If cyclo.HasCoefficentAbs(i) Then                    Console.WriteLine("CP[{0}] has coefficient with magnitude = {1}", n, i)                    n -= 1                    Exit While                End If            End While        Next    End Sub End Module`
Output:
```Task 1:  cyclotomic polynomials for n <= 30:
CP[1] = x - 1
CP[2] = x + 1
CP[3] = x^2 + x + 1
CP[4] = x^2 + 1
CP[5] = x^4 + x^3 + x^2 + x + 1
CP[6] = x^2 - x + 1
CP[7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[8] = x^4 + 1
CP[9] = x^6 + x^3 + 1
CP[10] = x^4 - x^3 + x^2 - x + 1
CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[12] = x^4 - x^2 + 1
CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
CP[16] = x^8 + 1
CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[18] = x^6 - x^3 + 1
CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[20] = x^8 - x^6 + x^4 - x^2 + 1
CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[24] = x^8 - x^4 + 1
CP[25] = x^20 + x^15 + x^10 + x^5 + 1
CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[27] = x^18 + x^9 + 1
CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:
CP[1] has coefficient with magnitude = 1
CP[105] has coefficient with magnitude = 2
CP[385] has coefficient with magnitude = 3
CP[1365] has coefficient with magnitude = 4
CP[1785] has coefficient with magnitude = 5
CP[2805] has coefficient with magnitude = 6
CP[3135] has coefficient with magnitude = 7
CP[6545] has coefficient with magnitude = 8
CP[6545] has coefficient with magnitude = 9
CP[10465] has coefficient with magnitude = 10```

## Wren

Translation of: Go
Library: Wren-trait
Library: Wren-sort
Library: Wren-math
Library: Wren-fmt

Second part is very slow. Limited to first 7 to finish in a reasonable time - 5 minutes on my machine.

`import "/trait" for Steppedimport "/sort" for Sortimport "/math" for Int, Numsimport "/fmt" for Fmt var algo = 2var maxAllFactors = 1e5 class Term {    construct new(coef, exp) {        _coef = coef        _exp = exp    }     coef { _coef }    exp  { _exp }     *(t) { Term.new(_coef * t.coef, _exp + t.exp) }     +(t) {        if (_exp != t.exp) Fiber.abort("Exponents unequal in term '+' method.")        return Term.new(_coef + t.coef, _exp)    }     - { Term.new(-_coef, _exp) }     toString {        if (_coef == 0) return "0"        if (_exp == 0)  return _coef.toString        if (_coef == 1) return (_exp == 1) ? "x" : "x^%(_exp)"        if (_exp == 1)  return "%(_coef)x"        return "%(_coef)x^%(_exp)"    }} class Poly {    // pass coef, exp in pairs as parameters    construct new(values) {        var le = values.count        if (le == 0) {            _terms = [Term.new(0, 0)]        } else {            if (le%2 != 0) Fiber.abort("Odd number of parameters(%(le)) passed to Poly constructor.")            _terms = []            for (i in Stepped.new(0...le, 2)) _terms.add(Term.new(values[i], values[i+1]))            tidy()        }    }     terms { _terms }     hasCoefAbs(coef) { _terms.any { |t| t.coef.abs == coef } }     +(p2) {        var p3 = Poly.new([])        var le = _terms.count        var le2 = p2.terms.count        while (le > 0 || le2 > 0) {            if (le == 0) {                p3.terms.add(p2.terms[le2-1])                le2 = le2 - 1            } else if (le2 == 0) {                p3.terms.add(_terms[le-1])                le = le - 1            } else {                var t = _terms[le-1]                var t2 = p2.terms[le2-1]                if (t.exp == t2.exp) {                    var t3 = t + t2                    if (t3.coef != 0) p3.terms.add(t3)                    le = le - 1                    le2 = le2 - 1                } else if (t.exp < t2.exp) {                    p3.terms.add(t)                    le = le - 1                } else {                    p3.terms.add(t2)                    le2 = le2 - 1                }            }        }        p3.tidy()        return p3    }     addTerm(t) {        var q = Poly.new([])        var added = false        for (i in 0..._terms.count) {            var ct = _terms[i]            if (ct.exp == t.exp) {                added = true                if (ct.coef + t.coef != 0) q.terms.add(ct + t)            } else {                q.terms.add(ct)            }        }        if (!added) q.terms.add(t)        q.tidy()        return q    }     mulTerm(t) {        var q = Poly.new([])        for (i in 0..._terms.count) {            var ct = _terms[i]            q.terms.add(ct * t)        }        q.tidy()        return q    }     /(v) {        var p = this        var q = Poly.new([])        var lcv = v.leadingCoef        var dv = v.degree        while (p.degree >= v.degree) {            var lcp = p.leadingCoef            var s = (lcp/lcv).truncate            var t = Term.new(s, p.degree - dv)            q = q.addTerm(t)            p = p + v.mulTerm(-t)         }        q.tidy()        return q    }     leadingCoef { _terms[0].coef }     degree { _terms[0].exp }     toString {        var sb = ""        var first = true        for (t in _terms) {            if (first) {                sb = sb + t.toString                first = false            } else {                sb = sb + " "                if (t.coef > 0) {                    sb = sb + "+ "                    sb = sb + t.toString                } else {                    sb = sb + "- "                    sb = sb + (-t).toString                }            }        }        return sb    }     // in place descending sort by term.exp    sortTerms() {        var cmp  = Fn.new { |t1, t2| (t2.exp - t1.exp).sign }        Sort.quick(_terms, 0, _terms.count-1, cmp)    }     // sort terms and remove any unnecesary zero terms    tidy() {        sortTerms()        if (degree > 0) {            for (i in _terms.count-1..0) {                if (_terms[i].coef == 0) _terms.removeAt(i)            }            if (_terms.count == 0) _terms.add(Term.new(0, 0))        }    }} var computed = {}var allFactors = {2: {2: 1}} var getFactors // recursive functiongetFactors = Fn.new { |n|    var f = allFactors[n]    if (f) return f    var factors = {}    if (n%2 == 0) {        var factorsDivTwo = getFactors.call(n/2)        for (me in factorsDivTwo) factors[me.key] = me.value        factors[2] = factors[2] ? factors[2] + 1 : 1        if (n < maxAllFactors) allFactors[n] = factors        return factors    }    var prime = true    var sqrt = n.sqrt.floor    var i = 3    while (i <= sqrt){        if (n%i == 0) {            prime = false            for (me in getFactors.call(n/i)) factors[me.key] = me.value            factors[i] = factors[i] ? factors[i] + 1 : 1            if (n < maxAllFactors) allFactors[n] = factors            return factors        }        i = i + 2    }    if (prime) {        factors[n] = 1        if (n < maxAllFactors) allFactors[n] = factors    }    return factors} var cycloPoly // recursive functioncycloPoly = Fn.new { |n|    var p = computed[n]    if (p) return p    if (n == 1) {        // polynomialL x - 1        p = Poly.new([1, 1, -1, 0])        computed[1] = p        return p    }    var factors = getFactors.call(n)    var cyclo = Poly.new([])    if (factors[n]) {        // n is prime        for (i in 0...n) cyclo.terms.add(Term.new(1, i))    } else if (factors.count == 2 && factors[2] == 1 && factors[n/2] == 1) {        // n == 2p        var prime = n / 2        var coef = -1        for (i in 0...prime) {            coef = coef * (-1)            cyclo.terms.add(Term.new(coef, i))        }    } else if (factors.count == 1) {        var h = factors[2]        if (h) { // n == 2^h            cyclo.terms.addAll([Term.new(1, 1 << (h-1)), Term.new(1, 0)])        } else if (!factors[n]) {            // n == p ^ k           var p = 0           for (prime in factors.keys) p = prime           var k = factors[p]           for (i in 0...p) {                var pk = p.pow(k-1).floor                cyclo.terms.add(Term.new(1, i * pk))           }        }    } else if (factors.count == 2 && factors[2]) {        // n = 2^h * p^k        var p = 0        for (prime in factors.keys) if (prime != 2) p = prime        var coef = -1        var twoExp = 1 << (factors[2] - 1)        var k = factors[p]        for (i in 0...p) {            coef = coef * (-1)            var pk = p.pow(k-1).floor            cyclo.terms.add(Term.new(coef, i * twoExp * pk))        }    } else if (factors[2] && (n/2) % 2 == 1 && (n/2) > 1) {        //  CP(2m)[x] == CP(-m)[x], n odd integer > 1        var cycloDiv2 = cycloPoly.call(n/2)        for (t in cycloDiv2.terms) {            var t2 = t            if (t.exp % 2 != 0) t2 = -t            cyclo.terms.add(t2)        }    } else if (algo == 0) {        // slow - uses basic definition        var divs = Int.properDivisors(n)        // polynomial: x^n - 1        var cyclo = Poly.new([1, n, -1, 0])        for (i in divs) {            var p = cycloPoly.call(i)            cyclo = cyclo / p        }    } else if (algo == 1) {        //  faster - remove max divisor (and all divisors of max divisor)        //  only one divide for all divisors of max divisor        var divs = Int.properDivisors(n)        var maxDiv = Nums.max(divs)        var divsExceptMax = divs.where { |d| maxDiv % d != 0 }.toList        // polynomial:  ( x^n - 1 ) / ( x^m - 1 ), where m is the max divisor        cyclo = Poly.new([1, n, -1, 0])        cyclo = cyclo / Poly.new([1, maxDiv, -1, 0])        for (i in divsExceptMax) {            var p = cycloPoly.call(i)            cyclo = cyclo / p        }    } else if (algo == 2) {        //  fastest        //  let p, q be primes such that p does not divide n, and q divides n        //  then CP(np)[x] = CP(n)[x^p] / CP(n)[x]        var m = 1        cyclo = cycloPoly.call(m)        var primes = []        for (prime in factors.keys) primes.add(prime)        Sort.quick(primes)        for (prime in primes) {            // CP(m)[x]            var cycloM = cyclo            // compute CP(m)[x^p]            var terms = []            for (t in cycloM.terms) terms.add(Term.new(t.coef, t.exp * prime))            cyclo = Poly.new([])            cyclo.terms.addAll(terms)            cyclo.tidy()            cyclo = cyclo / cycloM            m = m * prime        }        //  now, m is the largest square free divisor of n         var s = n / m        //  Compute CP(n)[x] = CP(m)[x^s]        var terms = []        for (t in cyclo.terms) terms.add(Term.new(t.coef, t.exp * s))        cyclo = Poly.new([])        cyclo.terms.addAll(terms)    } else {        Fiber.abort("Invalid algorithm.")    }    cyclo.tidy()    computed[n] = cyclo    return cyclo} System.print("Task 1:  cyclotomic polynomials for n <= 30:")for (i in 1..30) {    var p = cycloPoly.call(i)    Fmt.print("CP[\$2d] = \$s", i, p)} System.print("\nTask 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:")var n = 0for (i in 1..7) {     while(true) {        n = n + 1        var cyclo = cycloPoly.call(n)        if (cyclo.hasCoefAbs(i)) {            Fmt.print("CP[\$d] has coefficient with magnitude = \$d", n, i)            n = n - 1            break        }    }}`
Output:
```Task 1:  cyclotomic polynomials for n <= 30:
CP[ 1] = x - 1
CP[ 2] = x + 1
CP[ 3] = x^2 + x + 1
CP[ 4] = x^2 + 1
CP[ 5] = x^4 + x^3 + x^2 + x + 1
CP[ 6] = x^2 - x + 1
CP[ 7] = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[ 8] = x^4 + 1
CP[ 9] = x^6 + x^3 + 1
CP[10] = x^4 - x^3 + x^2 - x + 1
CP[11] = x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[12] = x^4 - x^2 + 1
CP[13] = x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[14] = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[15] = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
CP[16] = x^8 + 1
CP[17] = x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[18] = x^6 - x^3 + 1
CP[19] = x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[20] = x^8 - x^6 + x^4 - x^2 + 1
CP[21] = x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
CP[22] = x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[23] = x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[24] = x^8 - x^4 + 1
CP[25] = x^20 + x^15 + x^10 + x^5 + 1
CP[26] = x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
CP[27] = x^18 + x^9 + 1
CP[28] = x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
CP[29] = x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
CP[30] = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1

Task 2:  Smallest cyclotomic polynomial with n or -n as a coefficient:
CP[1] has coefficient with magnitude = 1
CP[105] has coefficient with magnitude = 2
CP[385] has coefficient with magnitude = 3
CP[1365] has coefficient with magnitude = 4
CP[1785] has coefficient with magnitude = 5
CP[2805] has coefficient with magnitude = 6
CP[3135] has coefficient with magnitude = 7
```