Fermat numbers

From Rosetta Code
Task
Fermat numbers
You are encouraged to solve this task according to the task description, using any language you may know.

In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form Fn = 22n + 1 where n is a non-negative integer.

Despite the simplicity of generating Fermat numbers, they have some powerful mathematical properties and are extensively used in cryptography & pseudo-random number generation, and are often linked to other number theoric fields.

As of this writing, (mid 2019), there are only five known prime Fermat numbers, the first five (F0 through F4). Only the first twelve Fermat numbers have been completely factored, though many have been partially factored.


Task
  • Write a routine (function, procedure, whatever) to generate Fermat numbers.
  • Use the routine to find and display here, on this page, the first 10 Fermat numbers - F0 through F9.
  • Find and display here, on this page, the prime factors of as many Fermat numbers as you have patience for. (Or as many as can be found in five minutes or less of processing time). Note: if you make it past F11, there may be money, and certainly will be acclaim in it for you.


See also


Arturo[edit]

nPowers: [1 2 4 8 16 32 64 128 256 512]
fermatSet: map 0..9 'x -> 1 + 2 ^ nPowers\[x]
 
loop 0..9 'i ->
    print ["F(" i ") =" fermatSet\[i]]

print ""

loop 0..9 'i ->
    print ["Prime factors of F(" i ") =" factors.prime fermatSet\[i]]

C[edit]

Compile with :
gcc -o fermat fermat.c -lgmp
#include <stdlib.h>
#include <stdio.h>
#include <gmp.h>

void mpz_factors(mpz_t n) {
  int factors = 0;
  mpz_t s, m, p;
  mpz_init(s), mpz_init(m), mpz_init(p);

  mpz_set_ui(m, 3);
  mpz_set(p, n);
  mpz_sqrt(s, p);

  while (mpz_cmp(m, s) < 0) {
    if (mpz_divisible_p(p, m)) {
      gmp_printf("%Zd ", m);
      mpz_fdiv_q(p, p, m);
      mpz_sqrt(s, p);
      factors ++;
    }
    mpz_add_ui(m, m, 2);
  }

  if (factors == 0) printf("PRIME\n");
  else gmp_printf("%Zd\n", p);
}

int main(int argc, char const *argv[]) {
  mpz_t fermat;
  mpz_init_set_ui(fermat, 3);
  printf("F(0) = 3 -> PRIME\n");
  for (unsigned i = 1; i < 10; i ++) {
    mpz_sub_ui(fermat, fermat, 1);
    mpz_mul(fermat, fermat, fermat);
    mpz_add_ui(fermat, fermat, 1);
    gmp_printf("F(%d) = %Zd -> ", i, fermat);
    mpz_factors(fermat);
  }

  return 0;
}
F(0) = 3 -> PRIME
F(1) = 5 -> PRIME
F(2) = 17 -> PRIME
F(3) = 257 -> PRIME
F(4) = 65537 -> PRIME
F(5) = 4294967297 -> 641 6700417
F(6) = 18446744073709551617 -> 274177 67280421310721
F(7) = 340282366920938463463374607431768211457 -> 59649589127497217 5704689200685129054721
F(8) = 115792089237316195423570985008687907853269984665640564039457584007913129639937 -> 1238926361552897 93461639715357977769163558199606896584051237541638188580280321
......

C++[edit]

Translation of: Java
Library: Boost

Built and tested on macOS 10.15, CPU: 3.2 GHz Intel Core i5. Execution time is about 12 minutes.

#include <iostream>
#include <vector>
#include <boost/integer/common_factor.hpp>
#include <boost/multiprecision/cpp_int.hpp>
#include <boost/multiprecision/miller_rabin.hpp>

typedef boost::multiprecision::cpp_int integer;

integer fermat(unsigned int n) {
    unsigned int p = 1;
    for (unsigned int i = 0; i < n; ++i)
        p *= 2;
    return 1 + pow(integer(2), p);
}

inline void g(integer& x, const integer& n) {
    x *= x;
    x += 1;
    x %= n;
}

integer pollard_rho(const integer& n) {
    integer x = 2, y = 2, d = 1, z = 1;
    int count = 0;
    for (;;) {
        g(x, n);
        g(y, n);
        g(y, n);
        d = abs(x - y);
        z = (z * d) % n;
        ++count;
        if (count == 100) {
            d = gcd(z, n);
            if (d != 1)
                break;
            z = 1;
            count = 0;
        }
    }
    if (d == n)
        return 0;
    return d;
}

std::vector<integer> get_prime_factors(integer n) {
    std::vector<integer> factors;
    for (;;) {
        if (miller_rabin_test(n, 25)) {
            factors.push_back(n);
            break;
        }
        integer f = pollard_rho(n);
        if (f == 0) {
            factors.push_back(n);
            break;
        }
        factors.push_back(f);
        n /= f;
    }
    return factors;
}

void print_vector(const std::vector<integer>& factors) {
    if (factors.empty())
        return;
    auto i = factors.begin();
    std::cout << *i++;
    for (; i != factors.end(); ++i)
        std::cout << ", " << *i;
    std::cout << '\n';
}

int main() {
    std::cout << "First 10 Fermat numbers:\n";
    for (unsigned int i = 0; i < 10; ++i)
        std::cout << "F(" << i << ") = " << fermat(i) << '\n';
    std::cout << "\nPrime factors:\n";
    for (unsigned int i = 0; i < 9; ++i) {
        std::cout << "F(" << i << "): ";
        print_vector(get_prime_factors(fermat(i)));
    }
    return 0;
}
Output:
First 10 Fermat numbers:
F(0) = 3
F(1) = 5
F(2) = 17
F(3) = 257
F(4) = 65537
F(5) = 4294967297
F(6) = 18446744073709551617
F(7) = 340282366920938463463374607431768211457
F(8) = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F(9) = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Prime factors:
F(0): 3
F(1): 5
F(2): 17
F(3): 257
F(4): 65537
F(5): 641, 6700417
F(6): 274177, 67280421310721
F(7): 59649589127497217, 5704689200685129054721
F(8): 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321


Common Lisp[edit]

Translation of: Lisp
This uses the 'factor' function defined in the page Prime Decomposition
http://rosettacode.org/wiki/Prime_decomposition#Common_Lisp
(defun fermat-number (n)
 "Return the n-th Fermat number"
  (1+ (expt 2 (expt 2 n))) )


(defun factor (n &optional (acc '()))
 "Return the list of factors of n"
  (when (> n 1) (loop with max-d = (isqrt n)
           for d = 2 then (if (evenp d) (1+ d) (+ d 2)) do
             (cond ((> d max-d) (return (cons (list n 1) acc)))
               ((zerop (rem n d)) 
                (return (factor (truncate n d) (if (eq d (caar acc))
                                   (cons 
                                (list (caar acc) (1+ (cadar acc)))
                                (cdr acc))
                                   (cons (list d 1) acc)))))))))
Output:
(dotimes (i 8) (format t "~d: ~d = ~d~%" i (fermat-number i) (factors (fermat-number i))))
0: 3 = (3)
1: 5 = (5)
2: 17 = (17)
3: 257 = (257)
4: 65537 = (65537)
5: 4294967297 = (641 6700417)
6: 18446744073709551617 = (274177 (67280421310721))
7: 340282366920938463463374607431768211457 = ((340282366920938463463374607431768211457))
8: 115792089237316195423570985008687907853269984665640564039457584007913129639937 = ((115792089237316195423570985008687907853269984665640564039457584007913129639937))
9: 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097 = ((13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097))
10: 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137217 = ((179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137217))
11: 32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638215525166389437335543602135433229604645318478604952148193555853611059596230657 = (319489
 (101152171346465785365739905563790778275446424351747584524444802254614885454171789617787158279386499891040749324458425859713854183244152133860909616251101863039512713637405344446131784663602352840930541077733717180487566766438342966931062084574042206368862921265008513729385286790910065162204496552694867070609361616173540692858549041708993328392702647150062512506151403207406283761595736673720405375033810606080158013948717662760215065784116654734290374983906448207065425365852408720566771005345821995341556493590254118091846659097349200248570452085641250044738949182704974520704370036542579394575574913724915713))
12: 1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085903001302413467189726673216491511131602920781738033436090243804708340403154190337 = (114689
 (9106268965752186405773463110818163752233991481723476361152625650968740750826648212547208641935996986118024454955917854702609434541985662158212523327009262247869952450049350838706079834460006786304075107567909269645531121898331250125751682239313156601738683820643686003638396435055834553570682260579462973839574318172464558815116581626749391315641251152532705571615644886981829338611134458123396450764186936496833100701185274214915961723337127995182593580031119299575446791424418154036863609858251201843852076223383379133238000289598800458955855329052103961332983048473420515918928565951506637819342706575976725030506905683310915700945442329953941604008255667676914945655757474715779252371155778495946746587469464160684843488975918662295274965457887082037460184558511575570318625886351712499453155527762335682281851520733417380809781252979478377941937578568481859702438295520231435016188495646093490407803983345420364088331996467459309353537828143080691834120737157445502646809195267166779721413577366833939771467773331873590129210913628329073978766992198221682739812652450408607796042492802295258713711959073218748776359806123717024800451461326745599716651128725627280643537507664130920416107218492950792456907321580171946770433))


Crystal[edit]

Translation of: Ruby
This uses the `factor` function from the `coreutils` library
that comes standard with most GNU/Linux, BSD, and Unix systems.
https://www.gnu.org/software/coreutils/
https://en.wikipedia.org/wiki/GNU_Core_Utilities
require "big"

def factors(n)
    factors = `factor #{n}`.split(' ')[1..-1].map(&.to_big_i)
    factors.group_by(&.itself).map { |prime, exp| [prime, exp.size] }
end

def fermat(n); (1.to_big_i << (1 << n)) | 1 end

puts "Value for each Fermat Number F0 .. F9."
(0..9).each { |n| puts "F#{n} = #{fermat(n)}" }
puts
puts "Factors for each Fermat Number F0 .. F8."
(0..8).each { |n| puts "F#{n} = #{factors fermat(n)}" }
System: Lenovo V570 (2011), I5-2410M, 2.9 GHz, Crystal 0.34
Run as: $ time crystal fermat.cr --release
Output:
Value for each Fermat Number F0 .. F9.
F0 = 3
F1 = 5
F2 = 17
F3 = 257
F4 = 65537
F5 = 4294967297
F6 = 18446744073709551617
F7 = 340282366920938463463374607431768211457
F8 = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F9 = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors for each Fermat Number F0 .. F8.
F0 = [[3, 1]]
F1 = [[5, 1]]
F2 = [[17, 1]]
F3 = [[257, 1]]
F4 = [[65537, 1]]
F5 = [[641, 1], [6700417, 1]]
F6 = [[274177, 1], [67280421310721, 1]]
F7 = [[59649589127497217, 1], [5704689200685129054721, 1]]
F8 = [[1238926361552897, 1], [93461639715357977769163558199606896584051237541638188580280321, 1]]
crystal fermat.cr --release  174.19s user 0.19s system 100% cpu 2:54.08 total

Factor[edit]

USING: formatting io kernel lists lists.lazy math math.functions
math.primes.factors sequences ;

: lfermats ( -- list )
    0 lfrom [ [ 1 2 2 ] dip ^ ^ + ] lmap-lazy ;

CHAR:  10 lfermats ltake list>array [
    "First 10 Fermat numbers:" print 
    [ dupd "F%c = %d\n" printf 1 + ] each drop nl
] [
    "Factors of first few Fermat numbers:" print [
        dupd factors dup length 1 = " (prime)" "" ?
        "Factors of F%c: %[%d, %]%s\n" printf 1 +
    ] each drop
] 2bi
Output:
First 10 Fermat numbers:
F₀ = 3
F₁ = 5
F₂ = 17
F₃ = 257
F₄ = 65537
F₅ = 4294967297
F₆ = 18446744073709551617
F₇ = 340282366920938463463374607431768211457
F₈ = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F₉ = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors of first few Fermat numbers:
Factors of F₀: { 3 } (prime)
Factors of F₁: { 5 } (prime)
Factors of F₂: { 17 } (prime)
Factors of F₃: { 257 } (prime)
Factors of F₄: { 65537 } (prime)
Factors of F₅: { 641, 6700417 }
Factors of F₆: { 274177, 67280421310721 }
^D

Go[edit]

The first seven Fermat numbers are factorized almost instantly by the Pollard's rho algorithm and, on switching to Lenstra's elliptical curve method, F₇ only takes a couple of seconds (compared to over 12 minutes when using Pollard's rho).

It's back to Pollard rho for F₈ and the first prime factor of F₉ which takes a further 40 seconds or so. ECM doesn't seem to be effective for numbers as big as these.

As the second and third prime factors of F₉ are respectively 49 and 99 digits long there would be no chance of finding these any time soon so I haven't bothered.

The ECM implementation is based on the Python code here.

The timings are for my Intel Core i7-8565U laptop using Go 1.14.1 on Ubuntu 18.04.

package main

import (
    "fmt"
    "github.com/jbarham/primegen"
    "math"
    "math/big"
    "math/rand"
    "sort"
    "time"
)

const (
    maxCurves = 10000
    maxRnd    = 1 << 31
    maxB1     = uint64(43 * 1e7)
    maxB2     = uint64(2 * 1e10)
)

var (
    zero  = big.NewInt(0)
    one   = big.NewInt(1)
    two   = big.NewInt(2)
    three = big.NewInt(3)
    four  = big.NewInt(4)
    five  = big.NewInt(5)
)

// Uses algorithm in Wikipedia article, including speed-up.
func pollardRho(n *big.Int) (*big.Int, error) {
    // g(x) = (x^2 + 1) mod n
    g := func(x, n *big.Int) *big.Int {
        x2 := new(big.Int)
        x2.Mul(x, x)
        x2.Add(x2, one)
        return x2.Mod(x2, n)
    }
    x, y, d := new(big.Int).Set(two), new(big.Int).Set(two), new(big.Int).Set(one)
    t, z := new(big.Int), new(big.Int).Set(one)
    count := 0
    for {
        x = g(x, n)
        y = g(g(y, n), n)
        t.Sub(x, y)
        t.Abs(t)
        t.Mod(t, n)
        z.Mul(z, t)
        count++
        if count == 100 {
            d.GCD(nil, nil, z, n)
            if d.Cmp(one) != 0 {
                break
            }
            z.Set(one)
            count = 0
        }
    }
    if d.Cmp(n) == 0 {
        return nil, fmt.Errorf("Pollard's rho failure")
    }
    return d, nil
}

// Gets all primes under 'n' - uses a Sieve of Atkin under the hood.
func getPrimes(n uint64) []uint64 {
    pg := primegen.New()
    var primes []uint64
    for {
        prime := pg.Next()
        if prime < n {
            primes = append(primes, prime)
        } else {
            break
        }
    }
    return primes
}

// Computes Stage 1 and Stage 2 bounds.
func computeBounds(n *big.Int) (uint64, uint64) {
    le := len(n.String())
    var b1, b2 uint64
    switch {
    case le <= 30:
        b1, b2 = 2000, 147396
    case le <= 40:
        b1, b2 = 11000, 1873422
    case le <= 50:
        b1, b2 = 50000, 12746592
    case le <= 60:
        b1, b2 = 250000, 128992510
    case le <= 70:
        b1, b2 = 1000000, 1045563762
    case le <= 80:
        b1, b2 = 3000000, 5706890290
    default:
        b1, b2 = maxB1, maxB2
    }
    return b1, b2
}

// Adds two specified P and Q points (in Montgomery form). Assumes R = P - Q.
func pointAdd(px, pz, qx, qz, rx, rz, n *big.Int) (*big.Int, *big.Int) {
    t := new(big.Int).Sub(px, pz)
    u := new(big.Int).Add(qx, qz)
    u.Mul(t, u)
    t.Add(px, pz)
    v := new(big.Int).Sub(qx, qz)
    v.Mul(t, v)
    upv := new(big.Int).Add(u, v)
    umv := new(big.Int).Sub(u, v)
    x := new(big.Int).Mul(upv, upv)
    x.Mul(x, rz)
    if x.Cmp(n) >= 0 {
        x.Mod(x, n)
    }
    z := new(big.Int).Mul(umv, umv)
    z.Mul(z, rx)
    if z.Cmp(n) >= 0 {
        z.Mod(z, n)
    }
    return x, z
}

// Doubles a point P (in Montgomery form).
func pointDouble(px, pz, n, a24 *big.Int) (*big.Int, *big.Int) {
    u2 := new(big.Int).Add(px, pz)
    u2.Mul(u2, u2)
    v2 := new(big.Int).Sub(px, pz)
    v2.Mul(v2, v2)
    t := new(big.Int).Sub(u2, v2)
    x := new(big.Int).Mul(u2, v2)
    if x.Cmp(n) >= 0 {
        x.Mod(x, n)
    }
    z := new(big.Int).Mul(a24, t)
    z.Add(v2, z)
    z.Mul(t, z)
    if z.Cmp(n) >= 0 {
        z.Mod(z, n)
    }
    return x, z
}

// Multiplies a specified point P (in Montgomery form) by a specified scalar.
func scalarMultiply(k, px, pz, n, a24 *big.Int) (*big.Int, *big.Int) {
    sk := fmt.Sprintf("%b", k)
    lk := len(sk)
    qx := new(big.Int).Set(px)
    qz := new(big.Int).Set(pz)
    rx, rz := pointDouble(px, pz, n, a24)
    for i := 1; i < lk; i++ {
        if sk[i] == '1' {
            qx, qz = pointAdd(rx, rz, qx, qz, px, pz, n)
            rx, rz = pointDouble(rx, rz, n, a24)

        } else {
            rx, rz = pointAdd(qx, qz, rx, rz, px, pz, n)
            qx, qz = pointDouble(qx, qz, n, a24)
        }
    }
    return qx, qz
}

// Lenstra's two-stage ECM algorithm.
func ecm(n *big.Int) (*big.Int, error) {
    if n.Cmp(one) == 0 || n.ProbablyPrime(10) {
        return n, nil
    }
    b1, b2 := computeBounds(n)
    dd := uint64(math.Sqrt(float64(b2)))
    beta := make([]*big.Int, dd+1)
    for i := 0; i < len(beta); i++ {
        beta[i] = new(big.Int)
    }
    s := make([]*big.Int, 2*dd+2)
    for i := 0; i < len(s); i++ {
        s[i] = new(big.Int)
    }

    // stage 1 and stage 2 precomputations
    curves := 0
    logB1 := math.Log(float64(b1))
    primes := getPrimes(b2)
    numPrimes := len(primes)
    idxB1 := sort.Search(len(primes), func(i int) bool { return primes[i] >= b1 })

    // compute a B1-powersmooth integer 'k'
    k := big.NewInt(1)
    for i := 0; i < idxB1; i++ {
        p := primes[i]
        bp := new(big.Int).SetUint64(p)
        t := uint64(logB1 / math.Log(float64(p)))
        bt := new(big.Int).SetUint64(t)
        bt.Exp(bp, bt, nil)
        k.Mul(k, bt)
    }
    g := big.NewInt(1)
    for (g.Cmp(one) == 0 || g.Cmp(n) == 0) && curves <= maxCurves {
        curves++
        st := int64(6 + rand.Intn(maxRnd-5))
        sigma := big.NewInt(st)

        // generate a new random curve in Montgomery form with Suyama's parameterization
        u := new(big.Int).Mul(sigma, sigma)
        u.Sub(u, five)
        u.Mod(u, n)
        v := new(big.Int).Mul(four, sigma)
        v.Mod(v, n)
        vmu := new(big.Int).Sub(v, u)
        a := new(big.Int).Mul(vmu, vmu)
        a.Mul(a, vmu)
        t := new(big.Int).Mul(three, u)
        t.Add(t, v)
        a.Mul(a, t)
        t.Mul(four, u)
        t.Mul(t, u)
        t.Mul(t, u)
        t.Mul(t, v)
        a.Quo(a, t)
        a.Sub(a, two)
        a.Mod(a, n)
        a24 := new(big.Int).Add(a, two)
        a24.Quo(a24, four)

        // stage 1
        px := new(big.Int).Mul(u, u)
        px.Mul(px, u)
        t.Mul(v, v)
        t.Mul(t, v)
        px.Quo(px, t)
        px.Mod(px, n)
        pz := big.NewInt(1)
        qx, qz := scalarMultiply(k, px, pz, n, a24)
        g.GCD(nil, nil, n, qz)

        // if stage 1 is successful, return a non-trivial factor else
        // move on to stage 2
        if g.Cmp(one) != 0 && g.Cmp(n) != 0 {
            return g, nil
        }

        // stage 2
        s[1], s[2] = pointDouble(qx, qz, n, a24)
        s[3], s[4] = pointDouble(s[1], s[2], n, a24)
        beta[1].Mul(s[1], s[2])
        beta[1].Mod(beta[1], n)
        beta[2].Mul(s[3], s[4])
        beta[2].Mod(beta[2], n)
        for d := uint64(3); d <= dd; d++ {
            d2 := 2 * d
            s[d2-1], s[d2] = pointAdd(s[d2-3], s[d2-2], s[1], s[2], s[d2-5], s[d2-4], n)
            beta[d].Mul(s[d2-1], s[d2])
            beta[d].Mod(beta[d], n)
        }
        g.SetUint64(1)
        b := new(big.Int).SetUint64(b1 - 1)
        rx, rz := scalarMultiply(b, qx, qz, n, a24)
        t.Mul(two, new(big.Int).SetUint64(dd))
        t.Sub(b, t)
        tx, tz := scalarMultiply(t, qx, qz, n, a24)
        q, step := idxB1, 2*dd
        for r := b1 - 1; r < b2; r += step {
            alpha := new(big.Int).Mul(rx, rz)
            alpha.Mod(alpha, n)
            limit := r + step
            for q < numPrimes && primes[q] <= limit {
                d := (primes[q] - r) / 2
                t := new(big.Int).Sub(rx, s[2*d-1])
                f := new(big.Int).Add(rz, s[2*d])
                f.Mul(t, f)
                f.Sub(f, alpha)
                f.Add(f, beta[d])
                g.Mul(g, f)
                g.Mod(g, n)
                q++
            }
            trx := new(big.Int).Set(rx)
            trz := new(big.Int).Set(rz)
            rx, rz = pointAdd(rx, rz, s[2*dd-1], s[2*dd], tx, tz, n)
            tx.Set(trx)
            tz.Set(trz)
        }
        g.GCD(nil, nil, n, g)
    }

    // no non-trivial factor found, return an error
    if curves > maxCurves {
        return zero, fmt.Errorf("maximum curves exceeded before a factor was found")
    }
    return g, nil
}

// find prime factors of 'n' using an appropriate method.
func primeFactors(n *big.Int) ([]*big.Int, error) {
    var res []*big.Int
    if n.ProbablyPrime(10) {
        return append(res, n), nil
    }
    le := len(n.String())
    var factor1 *big.Int
    var err error
    if le > 20 && le <= 60 {
        factor1, err = ecm(n)
    } else {
        factor1, err = pollardRho(n)
    }
    if err != nil {
        return nil, err
    }
    if !factor1.ProbablyPrime(10) {
        return nil, fmt.Errorf("first factor is not prime")
    }
    factor2 := new(big.Int)
    factor2.Quo(n, factor1)
    if !factor2.ProbablyPrime(10) {
        return nil, fmt.Errorf("%d (second factor is not prime)", factor1)
    }
    return append(res, factor1, factor2), nil
}

func fermatNumbers(n int) (res []*big.Int) {
    f := new(big.Int).SetUint64(3) // 2^1 + 1
    for i := 0; i < n; i++ {
        t := new(big.Int).Set(f)
        res = append(res, t)
        f.Sub(f, one)
        f.Mul(f, f)
        f.Add(f, one)
    }
    return res
}

func main() {
    start := time.Now()
    rand.Seed(time.Now().UnixNano())
    fns := fermatNumbers(10)
    fmt.Println("First 10 Fermat numbers:")
    for i, f := range fns {
        fmt.Printf("F%c = %d\n", 0x2080+i, f)
    }

    fmt.Println("\nFactors of first 10 Fermat numbers:")
    for i, f := range fns {
        fmt.Printf("F%c = ", 0x2080+i)
        factors, err := primeFactors(f)
        if err != nil {
            fmt.Println(err)
            continue
        }
        for _, factor := range factors {
            fmt.Printf("%d ", factor)
        }
        if len(factors) == 1 {
            fmt.Println("- prime")
        } else {
            fmt.Println()
        }
    }
    fmt.Printf("\nTook %s\n", time.Since(start))
}
Output:
First 10 Fermat numbers:
F₀ = 3
F₁ = 5
F₂ = 17
F₃ = 257
F₄ = 65537
F₅ = 4294967297
F₆ = 18446744073709551617
F₇ = 340282366920938463463374607431768211457
F₈ = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F₉ = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors of first 10 Fermat numbers:
F₀ = 3 - prime
F₁ = 5 - prime
F₂ = 17 - prime
F₃ = 257 - prime
F₄ = 65537 - prime
F₅ = 641 6700417 
F₆ = 274177 67280421310721 
F₇ = 59649589127497217 5704689200685129054721
F₈ = 1238926361552897 93461639715357977769163558199606896584051237541638188580280321 
F₉ = 2424833 (second factor is not prime)

Took 41.683532956s

Haskell[edit]

import Data.Numbers.Primes (primeFactors)
import Data.Bool (bool)

fermat :: Integer -> Integer
fermat = succ . (2 ^) . (2 ^)

fermats :: [Integer]
fermats = fermat <$> [0 ..]

--------------------------- TEST ---------------------------
main :: IO ()
main =
  mapM_
    putStrLn
    [ fTable "First 10 Fermats:" show show fermat [0 .. 9]
    , fTable
        "Factors of first 7:"
        show
        showFactors
        primeFactors
        (take 7 fermats)
    ]

------------------------- DISPLAY --------------------------
fTable :: String -> (a -> String) -> (b -> String) -> (a -> b) -> [a] -> String
fTable s xShow fxShow f xs =
  unlines $
  s : fmap (((++) . rjust w ' ' . xShow) <*> ((" -> " ++) . fxShow . f)) xs
  where
    rjust n c = drop . length <*> (replicate n c ++)
    w = maximum (length . xShow <$> xs)

showFactors :: [Integer] -> String
showFactors x
  | 1 < length x = show x
  | otherwise = "(prime)"
Output:
First 10 Fermats:
0 -> 3
1 -> 5
2 -> 17
3 -> 257
4 -> 65537
5 -> 4294967297
6 -> 18446744073709551617
7 -> 340282366920938463463374607431768211457
8 -> 115792089237316195423570985008687907853269984665640564039457584007913129639937
9 -> 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors of first 7:
                   3 -> (prime)
                   5 -> (prime)
                  17 -> (prime)
                 257 -> (prime)
               65537 -> (prime)
          4294967297 -> [641,6700417]
18446744073709551617 -> [274177,67280421310721]

J[edit]

   fermat =: 1 1 p. 2 ^ 2 ^ x:
   (,. fermat)i.10
0                                                                                                                                                           3
1                                                                                                                                                           5
2                                                                                                                                                          17
3                                                                                                                                                         257
4                                                                                                                                                       65537
5                                                                                                                                                  4294967297
6                                                                                                                                        18446744073709551617
7                                                                                                                     340282366920938463463374607431768211457
8                                                                              115792089237316195423570985008687907853269984665640564039457584007913129639937
9 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

   (; q:@:fermat)&>i.7
+-+---------------------+
|0|3                    |
+-+---------------------+
|1|5                    |
+-+---------------------+
|2|17                   |
+-+---------------------+
|3|257                  |
+-+---------------------+
|4|65537                |
+-+---------------------+
|5|641 6700417          |
+-+---------------------+
|6|274177 67280421310721|
+-+---------------------+

Java[edit]

import java.math.BigInteger;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.stream.Collectors;

public class FermatNumbers {

    public static void main(String[] args) {
        System.out.println("First 10 Fermat numbers:");
        for ( int i = 0 ; i < 10 ; i++ ) {
            System.out.printf("F[%d] = %s\n", i, fermat(i));
        }
        System.out.printf("%nFirst 12 Fermat numbers factored:%n");
        for ( int i = 0 ; i < 13 ; i++ ) {
            System.out.printf("F[%d] = %s\n", i, getString(getFactors(i, fermat(i))));
        }
    }
    
    private static String getString(List<BigInteger> factors) {
        if ( factors.size() == 1 ) {
            return factors.get(0) + " (PRIME)";
        }
        return factors.stream().map(v -> v.toString()).map(v -> v.startsWith("-") ? "(C" + v.replace("-", "") + ")" : v).collect(Collectors.joining(" * "));
    }

    private static Map<Integer, String> COMPOSITE = new HashMap<>();
    static {
        COMPOSITE.put(9, "5529");
        COMPOSITE.put(10, "6078");
        COMPOSITE.put(11, "1037");
        COMPOSITE.put(12, "5488");
        COMPOSITE.put(13, "2884");
    }

    private static List<BigInteger> getFactors(int fermatIndex, BigInteger n) {
        List<BigInteger> factors = new ArrayList<>();
        BigInteger factor = BigInteger.ONE;
        while ( true ) {
            if ( n.isProbablePrime(100) ) {
                factors.add(n);
                break;
            }
            else {
                if ( COMPOSITE.containsKey(fermatIndex) ) {
                    String stop = COMPOSITE.get(fermatIndex);
                    if ( n.toString().startsWith(stop) ) {
                        factors.add(new BigInteger("-" + n.toString().length()));
                        break;
                    }
                }
                factor = pollardRhoFast(n);
                if ( factor.compareTo(BigInteger.ZERO) == 0 ) {
                    factors.add(n);
                    break;
                }
                else {
                    factors.add(factor);
                    n = n.divide(factor);
                }
            }
        }
        return factors;
    }
    
    private static final BigInteger TWO = BigInteger.valueOf(2);
    
    private static BigInteger fermat(int n) {
        return TWO.pow((int)Math.pow(2, n)).add(BigInteger.ONE);
    }
        
    //  See:  https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
    @SuppressWarnings("unused")
    private static BigInteger pollardRho(BigInteger n) {
        BigInteger x = BigInteger.valueOf(2);
        BigInteger y = BigInteger.valueOf(2);
        BigInteger d = BigInteger.ONE;
        while ( d.compareTo(BigInteger.ONE) == 0 ) {
            x = pollardRhoG(x, n);
            y = pollardRhoG(pollardRhoG(y, n), n);
            d = x.subtract(y).abs().gcd(n);
        }
        if ( d.compareTo(n) == 0 ) {
            return BigInteger.ZERO;
        }
        return d;
    }
    
    //  Includes Speed Up of 100 multiples and 1 GCD, instead of 100 multiples and 100 GCDs.
    //  See Variants section of Wikipedia article.
    //  Testing F[8] = 1238926361552897 * Prime 
    //    This variant = 32 sec.
    //    Standard algorithm = 107 sec.
    private static BigInteger pollardRhoFast(BigInteger n) {
        long start = System.currentTimeMillis();
        BigInteger x = BigInteger.valueOf(2);
        BigInteger y = BigInteger.valueOf(2);
        BigInteger d = BigInteger.ONE;
        int count = 0;
        BigInteger z = BigInteger.ONE;
        while ( true ) {
            x = pollardRhoG(x, n);
            y = pollardRhoG(pollardRhoG(y, n), n);
            d = x.subtract(y).abs();
            z = z.multiply(d).mod(n);
            count++;
            if ( count == 100 ) {
                d = z.gcd(n);
                if ( d.compareTo(BigInteger.ONE) != 0 ) {
                    break;
                }
                z = BigInteger.ONE;
                count = 0;
            }
        }
        long end = System.currentTimeMillis();
        System.out.printf("    Pollard rho try factor %s elapsed time = %d ms (factor = %s).%n", n, (end-start), d);
        if ( d.compareTo(n) == 0 ) {
            return BigInteger.ZERO;
        }
        return d;
    }

    private static BigInteger pollardRhoG(BigInteger x, BigInteger n) {
        return x.multiply(x).add(BigInteger.ONE).mod(n);
    }

}

Output includes composite numbers attempted to factor with Pollard rho.

Output:
First 10 Fermat numbers:
F[0] = 3
F[1] = 5
F[2] = 17
F[3] = 257
F[4] = 65537
F[5] = 4294967297
F[6] = 18446744073709551617
F[7] = 340282366920938463463374607431768211457
F[8] = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F[9] = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

First 12 Fermat numbers factored:
F[0] = 3 (PRIME)
F[1] = 5 (PRIME)
F[2] = 17 (PRIME)
F[3] = 257 (PRIME)
F[4] = 65537 (PRIME)
    Pollard rho try factor 4294967297 elapsed time = 2 ms (factor = 641).
F[5] = 641 * 6700417
    Pollard rho try factor 18446744073709551617 elapsed time = 6 ms (factor = 274177).
F[6] = 274177 * 67280421310721
    Pollard rho try factor 340282366920938463463374607431768211457 elapsed time = 574251 ms (factor = 59649589127497217).
F[7] = 59649589127497217 * 5704689200685129054721
    Pollard rho try factor 115792089237316195423570985008687907853269984665640564039457584007913129639937 elapsed time = 31640 ms (factor = 1238926361552897).
F[8] = 1238926361552897 * 93461639715357977769163558199606896584051237541638188580280321
    Pollard rho try factor 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097 elapsed time = 30 ms (factor = 2424833).
F[9] = 2424833 * (C148)
    Pollard rho try factor 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137217 elapsed time = 120 ms (factor = 45592577).
    Pollard rho try factor 3942951359960012586542991835686376608231592127249807732373409846031135195659174148737161255930050543559319182152642816343958573976075461198274610155058226350701077796608546283231637018483208223116080561800334422176622099740983337736621316898600121619871377542107047343253864459964167331555646795960321 elapsed time = 5026 ms (factor = 6487031809).
F[10] = 45592577 * 6487031809 * (C291)
    Pollard rho try factor 32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638215525166389437335543602135433229604645318478604952148193555853611059596230657 elapsed time = 8 ms (factor = 974849).
    Pollard rho try factor 33150781373639412155846573868024639672856106606987835072026893834352453701925006737655987144186344206834820532125383540932102878651453631377873037143648178457002958783669056532601662155256508553423204658756451069116132055982639112479817996775373591674794399801442382402697828988429044712163168243619196804348072710121945390948428910309765481110260333687910970886853046635254307274981520537180895290310783635953818082306553996934497908037349010876970379631341148456045116407475229712217130141926525362871253437794629422541384355185626695660779797862427347553871011957167960991543632376506466281643163047416635393 elapsed time = 98 ms (factor = 319489).
F[11] = 974849 * 319489 * (C606)
    Pollard rho try factor 1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085903001302413467189726673216491511131602920781738033436090243804708340403154190337 elapsed time = 75 ms (factor = 114689).
    Pollard rho try factor 9106268965752186405773463110818163752233991481723476361152625650968740750826648212547208641935996986118024454955917854702609434541985662158212523327009262247869952450049350838706079834460006786304075107567909269645531121898331250125751682239313156601738683820643686003638396435055834553570682260579462973839574318172464558815116581626749391315641251152532705571615644886981829338611134458123396450764186936496833100701185274214915961723337127995182593580031119299575446791424418154036863609858251201843852076223383379133238000289598800458955855329052103961332983048473420515918928565951506637819342706575976725030506905683310915700945442329953941604008255667676914945655757474715779252371155778495946746587469464160684843488975918662295274965457887082037460184558511575570318625886351712499453155527762335682281851520733417380809781252979478377941937578568481859702438295520231435016188495646093490407803983345420364088331996467459309353537828143080691834120737157445502646809195267166779721413577366833939771467773331873590129210913628329073978766992198221682739812652450408607796042492802295258713711959073218748776359806123717024800451461326745599716651128725627280643537507664130920416107218492950792456907321580171946770433 elapsed time = 301 ms (factor = 26017793).
    Pollard rho try factor 350001591824186871106763863899530746217943677302816436473017740242946077356624684213115564488348300185108877411543625345263121839042445381828217455916005721464151569276047005167043946981206545317048033534893189043572263100806868980998952610596646556521480658280419327021257968923645235918768446677220584153297488306270426504473941414890547838382804073832577020334339845312084285496895699728389585187497914849919557000902623608963141559444997044721763816786117073787751589515083702681348245404913906488680729999788351730419178916889637812821243344085799435845038164784900107242721493170135785069061880328434342030106354742821995535937481028077744396075726164309052460585559946407282864168038994640934638329525854255227752926198464207255983432778402344965903148839661825873175277621985527846249416909718758069997783882773355041329208102046913755441975327368023946523920699020098723785533557579080342841062805878477869513695185309048285123705067072486920463781103076554014502567884803571416673251784936825115787932810954867447447568320403976197134736485611912650805539603318790667901618038578533362100071745480995207732506742832634459994375828162163700807237997808869771569154136465922798310222055287047244647419069003284481 elapsed time = 1616 ms (factor = 63766529).
F[12] = 114689 * 26017793 * 63766529 * (C1213)

jq[edit]

Works with gojq, the Go implementation of jq

Preliminaries

# To take advantage of gojq's arbitrary-precision integer arithmetic:
def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);

def gcd(a; b):
  # subfunction expects [a,b] as input
  # i.e. a ~ .[0] and b ~ .[1]
  def rgcd: if .[1] == 0 then .[0]
         else [.[1], .[0] % .[1]] | rgcd
         end;
  [a,b] | rgcd;

# This is fast because the state of `until` is just a number
def is_prime:
  . as $n
  | if ($n < 2)         then false
    elif ($n % 2 == 0)  then $n == 2
    elif ($n % 3 == 0)  then $n == 3
    elif ($n % 5 == 0)  then $n == 5
    elif ($n % 7 == 0)  then $n == 7
    elif ($n % 11 == 0) then $n == 11
    elif ($n % 13 == 0) then $n == 13
    elif ($n % 17 == 0) then $n == 17
    elif ($n % 19 == 0) then $n == 19
    elif ($n % 23 == 0) then $n == 23
    elif ($n % 29 == 0) then $n == 29
    elif ($n % 31 == 0) then $n == 31
    elif ($n % 37 == 0) then $n == 37
    elif ($n % 41 == 0) then $n == 41
    else 43
    | until( (. * .) > $n or ($n % . == 0); . + 2)
    | . * . > $n
    end;

Fermat Numbers

def fermat:
  . as $n
  | (2 | power( 2 | power($n))) + 1;

# https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
def pollardRho($x):
  . as $n
  | def g: (.*. + 1) % $n ;
  {x:$x, y:$x, d:1}
  | until(.d != 1;
         .x |= g
       | .y |= (g|g)
       | .d = gcd((.x - .y)|length; $n) )
  | if .d == $n then 0 
    else .d
    end ;

def rhoPrimeFactors:
  . as $n
  | pollardRho(2)
  | if . == 0
    then [$n, 1]
    else [., ($n / .)]
    end ;
 
"The first 10 Fermat numbers are:",
 [ range(0;10) | fermat ] as $fns
 | (range(0;10) | "F\(.) is \($fns[.])"),
 
   ("\nFactors of the first 7 Fermat numbers:",
    range(0;7) as $i
    | $fns[$i]
    | rhoPrimeFactors as $factors
    | if $factors[1] == 1
      then "F\($i) : rho-prime", " ... => \(if is_prime then "prime" else "not" end)"
      else "F\($i) => \($factors)"
      end )
Output:
The first 10 Fermat numbers are:
F0 is 3
F1 is 5
F2 is 17
F3 is 257
F4 is 65537
F5 is 4294967297
F6 is 18446744073709551617
F7 is 340282366920938463463374607431768211457
F8 is 115792089237316195423570985008687907853269984665640564039457584007913129639937
F9 is 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors of the first 7 Fermat numbers:
F0 : rho-prime
 ... => prime
F1 : rho-prime
 ... => prime
F2 : rho-prime
 ... => prime
F3 : rho-prime
 ... => prime
F4 : rho-prime
 ... => prime
F5 => [641,6700417]
F6 => [274177,67280421310721]


Julia[edit]

using Primes

fermat(n) = BigInt(2)^(BigInt(2)^n) + 1
prettyprint(fdict) = replace(replace(string(fdict), r".+\(([^)]+)\)" => s"\1"), r"\=\>" => "^")

function factorfermats(max, nofactor=false) 
    for n in 0:max
        fm = fermat(n)
        if nofactor
            println("Fermat number F($n) is $fm.")
            continue
        end
        factors = factor(fm)
        println("Fermat number F($n), $fm, ", 
            length(factors) < 2 ? "is prime." : "factors to $(prettyprint(factors)).")
    end
end

factorfermats(9, true)
factorfermats(10)
Output:
Fermat number F(0) is 3.
Fermat number F(1) is 5.
Fermat number F(2) is 17.
Fermat number F(3) is 257.
Fermat number F(4) is 65537.
Fermat number F(5) is 4294967297.
Fermat number F(6) is 18446744073709551617.
Fermat number F(7) is 340282366920938463463374607431768211457.
Fermat number F(8) is 115792089237316195423570985008687907853269984665640564039457584007913129639937.
Fermat number F(9) is 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097.
Fermat number F(0), 3, is prime.
Fermat number F(1), 5, is prime.
Fermat number F(2), 17, is prime.
Fermat number F(3), 257, is prime.
Fermat number F(4), 65537, is prime.
Fermat number F(5), 4294967297, factors to 641^1,6700417^1.
Fermat number F(6), 18446744073709551617, factors to 274177^1,67280421310721^1.
...waited >5 minutes

Kotlin[edit]

Translation of: Java
import java.math.BigInteger
import kotlin.math.pow

fun main() {
    println("First 10 Fermat numbers:")
    for (i in 0..9) {
        println("F[$i] = ${fermat(i)}")
    }
    println()
    println("First 12 Fermat numbers factored:")
    for (i in 0..12) {
        println("F[$i] = ${getString(getFactors(i, fermat(i)))}")
    }
}

private fun getString(factors: List<BigInteger>): String {
    return if (factors.size == 1) {
        "${factors[0]} (PRIME)"
    } else factors.map { it.toString() }
        .joinToString(" * ") {
            if (it.startsWith("-"))
                "(C" + it.replace("-", "") + ")"
            else it
        }
}

private val COMPOSITE = mutableMapOf(
    9 to "5529",
    10 to "6078",
    11 to "1037",
    12 to "5488",
    13 to "2884"
)

private fun getFactors(fermatIndex: Int, n: BigInteger): List<BigInteger> {
    var n2 = n
    val factors: MutableList<BigInteger> = ArrayList()
    var factor: BigInteger
    while (true) {
        if (n2.isProbablePrime(100)) {
            factors.add(n2)
            break
        } else {
            if (COMPOSITE.containsKey(fermatIndex)) {
                val stop = COMPOSITE[fermatIndex]
                if (n2.toString().startsWith(stop!!)) {
                    factors.add(BigInteger("-" + n2.toString().length))
                    break
                }
            }
            //factor = pollardRho(n)
            factor = pollardRhoFast(n)
            n2 = if (factor.compareTo(BigInteger.ZERO) == 0) {
                factors.add(n2)
                break
            } else {
                factors.add(factor)
                n2.divide(factor)
            }
        }
    }
    return factors
}

private val TWO = BigInteger.valueOf(2)
private fun fermat(n: Int): BigInteger {
    return TWO.pow(2.0.pow(n.toDouble()).toInt()).add(BigInteger.ONE)
}

//  See:  https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
@Suppress("unused")
private fun pollardRho(n: BigInteger): BigInteger {
    var x = BigInteger.valueOf(2)
    var y = BigInteger.valueOf(2)
    var d = BigInteger.ONE
    while (d.compareTo(BigInteger.ONE) == 0) {
        x = pollardRhoG(x, n)
        y = pollardRhoG(pollardRhoG(y, n), n)
        d = (x - y).abs().gcd(n)
    }
    return if (d.compareTo(n) == 0) {
        BigInteger.ZERO
    } else d
}

//  Includes Speed Up of 100 multiples and 1 GCD, instead of 100 multiples and 100 GCDs.
//  See Variants section of Wikipedia article.
//  Testing F[8] = 1238926361552897 * Prime
//    This variant = 32 sec.
//    Standard algorithm = 107 sec.
private fun pollardRhoFast(n: BigInteger): BigInteger {
    val start = System.currentTimeMillis()
    var x = BigInteger.valueOf(2)
    var y = BigInteger.valueOf(2)
    var d: BigInteger
    var count = 0
    var z = BigInteger.ONE
    while (true) {
        x = pollardRhoG(x, n)
        y = pollardRhoG(pollardRhoG(y, n), n)
        d = (x - y).abs()
        z = (z * d).mod(n)
        count++
        if (count == 100) {
            d = z.gcd(n)
            if (d.compareTo(BigInteger.ONE) != 0) {
                break
            }
            z = BigInteger.ONE
            count = 0
        }
    }
    val end = System.currentTimeMillis()
    println("    Pollard rho try factor $n elapsed time = ${end - start} ms (factor = $d).")
    return if (d.compareTo(n) == 0) {
        BigInteger.ZERO
    } else d
}

private fun pollardRhoG(x: BigInteger, n: BigInteger): BigInteger {
    return (x * x + BigInteger.ONE).mod(n)
}

langur[edit]

Translation of: Python
Works with: langur version 0.10
val .fermat = f 2 ^ 2 ^ .n + 1

val .factors = f(var .x) {
    for[.f=[]] .i, .s = 2, truncate .x ^/ 2; .i < .s; .i += 1 {
        if .x div .i {
            .f ~= [.i]
            .x \= .i
            .s = truncate .x ^/ 2
        }
    } ~ [.x]
}

writeln "first 10 Fermat numbers"
for .i in 0..9 {
    writeln $"F\(.i + 16x2080:cp) = \(.fermat(.i))"
}
writeln()

writeln "factors of first few Fermat numbers"
for .i in 0..9 {
    val .ferm = .fermat(.i)
    val .fac = .factors(.ferm)
    if len(.fac) == 1 {
        writeln $"F\(.i + 16x2080:cp) is prime"
    } else {
        writeln $"F\(.i + 16x2080:cp) factors: ", .fac
    }
}
Output:

I just ran an initial test. Maybe I'll take the time to calculate more factors later.

first 10 Fermat numbers
F₀ = 3
F₁ = 5
F₂ = 17
F₃ = 257
F₄ = 65537
F₅ = 4294967297
F₆ = 18446744073709551617
F₇ = 340282366920938463463374607431768211457
F₈ = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F₉ = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

factors of first few Fermat numbers
F₀ is prime
F₁ is prime
F₂ is prime
F₃ is prime
F₄ is prime
F₅ factors: [641, 6700417]
F₆ factors: [274177, 67280421310721]

Mathematica / Wolfram Language[edit]

ClearAll[Fermat]
Fermat[n_] := 2^(2^n) + 1
Fermat /@ Range[0, 9]
Scan[FactorInteger /* Print, %]
Output:
{3,5,17,257,65537,4294967297,18446744073709551617,340282366920938463463374607431768211457,115792089237316195423570985008687907853269984665640564039457584007913129639937,13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097}
{{3,1}}
{{5,1}}
{{17,1}}
{{257,1}}
{{65537,1}}
{{641,1},{6700417,1}}
{{274177,1},{67280421310721,1}}
{{59649589127497217,1},{5704689200685129054721,1}}
{{1238926361552897,1},{93461639715357977769163558199606896584051237541638188580280321,1}}
$Aborted

Nim[edit]

Translation of: Kotlin
Library: bignum
import math
import bignum
import strformat
import strutils
import tables
import times

const Composite = {9: "5529", 10: "6078", 11: "1037", 12: "5488", 13: "2884"}.toTable

const Subscripts = ["₀", "₁", "₂", "₃", "₄", "₅", "₆", "₇", "₈", "₉"]

let One = newInt(1)

#---------------------------------------------------------------------------------------------------

func fermat(n: int): Int {.inline.} = 2^(culong(2^n)) + 1

#---------------------------------------------------------------------------------------------------

template isProbablyPrime(n: Int): bool = n.probablyPrime(25) != 0

#---------------------------------------------------------------------------------------------------

func pollardRhoG(x, n: Int): Int {.inline.} = (x * x + 1) mod n

#---------------------------------------------------------------------------------------------------

proc pollardRhoFast(n: Int): Int =

  let start = getTime()
  var
    x = newInt(2)
    y = newInt(2)
    count = 0
    z = One

  while true:
    x = pollardRhoG(x, n)
    y = pollardRhoG(pollardRhoG(y, n), n)
    result = abs(x - y)
    z = z * result mod n
    inc count
    if count == 100:
      result = gcd(z, n)
      if result != One: break
      z = One
      count = 0

  let duration = (getTime() - start).inMilliseconds
  echo fmt"    Pollard rho try factor {n} elapsed time = {duration} ms (factor = {result})."
  if result == n:
    result = newInt(0)

#---------------------------------------------------------------------------------------------------

proc factors(fermatIndex: int; n: Int): seq[Int] =

  var n = n
  var factor: Int
  while true:

    if n.isProbablyPrime():
      result.add(n)
      break

    if fermatIndex in Composite:
      let stop = Composite[fermatIndex]
      let s = $n
      if s.startsWith(stop):
        result.add(newInt(-s.len))
        break

    factor = pollardRhoFast(n)
    if factor.isZero():
      result.add(n)
      break
    result.add(factor)
    n = n div factor

#---------------------------------------------------------------------------------------------------

func `$`(factors: seq[Int]): string =

  if factors.len == 1:
    result = fmt"{factors[0]} (PRIME)"

  else:
    result = $factors[0]
    let start = result.high
    for factor in factors[1..^1]:
      result.addSep(" * ", start)
      result.add(if factor < 0: fmt"(C{-factor})" else: $factor)

#---------------------------------------------------------------------------------------------------

func subscript(n: Natural): string =
  var n = n
  while true:
    result.insert(Subscripts[n mod 10], 0)
    n = n div 10
    if n == 0: break

#———————————————————————————————————————————————————————————————————————————————————————————————————

echo "First 10 Fermat numbers:"
for i in 0..9:
  echo fmt"F{subscript(i)} = {fermat(i)}"

echo ""
echo "First 12 Fermat numbers factored:"
for i in 0..12:
  echo fmt"F{subscript(i)} = {factors(i, fermat(i))}"
Output:
First 10 Fermat numbers:
F₀ = 3
F₁ = 5
F₂ = 17
F₃ = 257
F₄ = 65537
F₅ = 4294967297
F₆ = 18446744073709551617
F₇ = 340282366920938463463374607431768211457
F₈ = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F₉ = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

First 12 Fermat numbers factored:
F₀ = 3 (PRIME)
F₁ = 5 (PRIME)
F₂ = 17 (PRIME)
F₃ = 257 (PRIME)
F₄ = 65537 (PRIME)
    Pollard rho try factor 4294967297 elapsed time = 0 ms (factor = 641).
F₅ = 641 * 6700417
    Pollard rho try factor 18446744073709551617 elapsed time = 1 ms (factor = 274177).
F₆ = 274177 * 67280421310721
    Pollard rho try factor 340282366920938463463374607431768211457 elapsed time = 516705 ms (factor = 59649589127497217).
F₇ = 59649589127497217 * 5704689200685129054721
    Pollard rho try factor 115792089237316195423570985008687907853269984665640564039457584007913129639937 elapsed time = 20765 ms (factor = 1238926361552897).
F₈ = 1238926361552897 * 93461639715357977769163558199606896584051237541638188580280321
    Pollard rho try factor 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097 elapsed time = 3 ms (factor = 2424833).
F₉ = 2424833 * (C148)
    Pollard rho try factor 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137217 elapsed time = 46 ms (factor = 45592577).
    Pollard rho try factor 3942951359960012586542991835686376608231592127249807732373409846031135195659174148737161255930050543559319182152642816343958573976075461198274610155058226350701077796608546283231637018483208223116080561800334422176622099740983337736621316898600121619871377542107047343253864459964167331555646795960321 elapsed time = 595 ms (factor = 6487031809).
F₁₀ = 45592577 * 6487031809 * (C291)
    Pollard rho try factor 32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638215525166389437335543602135433229604645318478604952148193555853611059596230657 elapsed time = 1 ms (factor = 974849).
    Pollard rho try factor 33150781373639412155846573868024639672856106606987835072026893834352453701925006737655987144186344206834820532125383540932102878651453631377873037143648178457002958783669056532601662155256508553423204658756451069116132055982639112479817996775373591674794399801442382402697828988429044712163168243619196804348072710121945390948428910309765481110260333687910970886853046635254307274981520537180895290310783635953818082306553996934497908037349010876970379631341148456045116407475229712217130141926525362871253437794629422541384355185626695660779797862427347553871011957167960991543632376506466281643163047416635393 elapsed time = 11 ms (factor = 319489).
F₁₁ = 974849 * 319489 * (C606)
    Pollard rho try factor 1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085903001302413467189726673216491511131602920781738033436090243804708340403154190337 elapsed time = 11 ms (factor = 114689).
    Pollard rho try factor 9106268965752186405773463110818163752233991481723476361152625650968740750826648212547208641935996986118024454955917854702609434541985662158212523327009262247869952450049350838706079834460006786304075107567909269645531121898331250125751682239313156601738683820643686003638396435055834553570682260579462973839574318172464558815116581626749391315641251152532705571615644886981829338611134458123396450764186936496833100701185274214915961723337127995182593580031119299575446791424418154036863609858251201843852076223383379133238000289598800458955855329052103961332983048473420515918928565951506637819342706575976725030506905683310915700945442329953941604008255667676914945655757474715779252371155778495946746587469464160684843488975918662295274965457887082037460184558511575570318625886351712499453155527762335682281851520733417380809781252979478377941937578568481859702438295520231435016188495646093490407803983345420364088331996467459309353537828143080691834120737157445502646809195267166779721413577366833939771467773331873590129210913628329073978766992198221682739812652450408607796042492802295258713711959073218748776359806123717024800451461326745599716651128725627280643537507664130920416107218492950792456907321580171946770433 elapsed time = 18 ms (factor = 26017793).
    Pollard rho try factor 350001591824186871106763863899530746217943677302816436473017740242946077356624684213115564488348300185108877411543625345263121839042445381828217455916005721464151569276047005167043946981206545317048033534893189043572263100806868980998952610596646556521480658280419327021257968923645235918768446677220584153297488306270426504473941414890547838382804073832577020334339845312084285496895699728389585187497914849919557000902623608963141559444997044721763816786117073787751589515083702681348245404913906488680729999788351730419178916889637812821243344085799435845038164784900107242721493170135785069061880328434342030106354742821995535937481028077744396075726164309052460585559946407282864168038994640934638329525854255227752926198464207255983432778402344965903148839661825873175277621985527846249416909718758069997783882773355041329208102046913755441975327368023946523920699020098723785533557579080342841062805878477869513695185309048285123705067072486920463781103076554014502567884803571416673251784936825115787932810954867447447568320403976197134736485611912650805539603318790667901618038578533362100071745480995207732506742832634459994375828162163700807237997808869771569154136465922798310222055287047244647419069003284481 elapsed time = 114 ms (factor = 63766529).
F₁₂ = 114689 * 26017793 * 63766529 * (C1213)

Perl[edit]

Library: ntheory
Translation of: Raku
use strict;
use warnings;
use feature 'say';
use bigint try=>"GMP";
use ntheory qw<factor>;

my @Fermats = map { 2**(2**$_) + 1 } 0..9;

my $sub = 0;
say 'First 10 Fermat numbers:';
printf "F%s = %s\n", $sub++, $_ for @Fermats;

$sub = 0;
say "\nFactors of first few Fermat numbers:";
for my $f (map { [factor($_)] } @Fermats[0..8]) {
   printf "Factors of F%s: %s\n", $sub++, @$f == 1 ? 'prime' : join ' ', @$f
}
Output:
First 10 Fermat numbers:
F0 = 3
F1 = 5
F2 = 17
F3 = 257
F4 = 65537
F5 = 4294967297
F6 = 18446744073709551617
F7 = 340282366920938463463374607431768211457
F8 = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F9 = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors of first few Fermat numbers:
Factors of F0: prime
Factors of F1: prime
Factors of F2: prime
Factors of F3: prime
Factors of F4: prime
Factors of F5: 641 6700417
Factors of F6: 274177 67280421310721
Factors of F7: 59649589127497217 5704689200685129054721
Factors of F8: 1238926361552897 93461639715357977769163558199606896584051237541638188580280321

Phix[edit]

Library: Phix/mpfr
with javascript_semantics
-- demo\rosetta\Fermat.exw
include mpfr.e
 
procedure fermat(mpz res, integer n)
    integer pn = power(2,n)
    mpz_ui_pow_ui(res,2,pn)
    mpz_add_si(res,res,1)
end procedure
 
mpz fn = mpz_init()
constant lim = iff(platform()=JS?18:29), -- (see note)
         print_lim = iff(platform()=JS?16:20)
for i=0 to lim do
    fermat(fn,i)
    if i<=print_lim then
        printf(1,"F%d = %s\n",{i,shorten(mpz_get_str(fn))})
    else -- (since printing it takes too long...)
        printf(1,"F%d has %,d digits\n",{i,mpz_sizeinbase(fn,10)})
    end if
end for
 
printf(1,"\n")
constant flimit = iff(platform()=JS?11:13)
for i=0 to flimit do
    atom t = time()
    fermat(fn,i)
    sequence f = mpz_prime_factors(fn, 200000)
    t = time()-t
    string fs = "",
           ts = elapsed(t)
    if length(f[$])=1 then -- (as per docs)
        mpz_set_str(fn,f[$][1])
        if not mpz_prime(fn) then
            if length(f)=1 then
                fs = " (not prime)"
            else
                fs = " (last factor is not prime)"
            end if
        end if
        f = deep_copy(f)
        f[$][1] = shorten(f[$][1])
    elsif length(f)=1
      and mpz_prime(fn) then
        fs = " (prime)"
    end if
    fs = mpz_factorstring(f)&fs
    printf(1,"Factors of F%d: %s [%s]\n",{i,fs,ts})
end for
Output:

Note that mpz_prime_factors(), a phix-specific extension to gmp, is designed to find small factors quickly and give up early, however it works by maintaining a table of primes, so any prime factor over 10 digits or so is beyond reach. You could increase the maxprime parameter, here set at 200,000, which guarantees all factors up to 2,750,159 (obviously 7 digits), but it will just get exponentially slower without getting close to finding anything more, specifically in this case 1,238,926,361,552,897 (16 digits) or 59,649,589,127,497,217 (17 digits).

Calculating F0..F29 is pretty quick, but F30 and above hit integer limits on 32 bit, F32 and above exceed my physical memory on 64 bit.
As noted above, there is not really much point, and it just takes far too long to bother printing out any numbers with more than 500,000 digits.
Attempting to factor F14 and above gets nowhere, with each attempt some 5-10 times slower than the previous, until F18 which eventually crashes.

F0 = 3
F1 = 5
F2 = 17
F3 = 257
F4 = 65537
F5 = 4294967297
F6 = 18446744073709551617
F7 = 340282366920938463463374607431768211457
F8 = 115792089...<78 digits>...129639937
F9 = 134078079...<155 digits>...006084097
F10 = 179769313...<309 digits>...224137217
F11 = 323170060...<617 digits>...596230657
F12 = 104438888...<1,234 digits>...154190337
F13 = 109074813...<2,467 digits>...715792897
F14 = 118973149...<4,933 digits>...964066817
F15 = 141546103...<9,865 digits>...712377857
F16 = 200352993...<19,729 digits>...719156737
F17 = 401413218...<39,457 digits>...934173697
F18 = 161132571...<78,914 digits>...298300417
F19 = 259637056...<157,827 digits>...185773057
F20 = 674114012...<315,653 digits>...335579137
F21 has 631,306 digits
F22 has 1,262,612 digits
F23 has 2,525,223 digits
F24 has 5,050,446 digits
F25 has 10,100,891 digits
F26 has 20,201,782 digits
F27 has 40,403,563 digits
F28 has 80,807,125 digits
F29 has 161,614,249 digits

Factors of F0: 3 (prime) [0.0s]
Factors of F1: 5 (prime) [0s]
Factors of F2: 17 (prime) [0s]
Factors of F3: 257 (prime) [0s]
Factors of F4: 65537 (prime) [0s]
Factors of F5: 641*6700417 [0s]
Factors of F6: 274177*67280421310721 [0.0s]
Factors of F7: 340282366920938463463374607431768211457 (not prime) [0.2s]
Factors of F8: 115792089...<78 digits>...129639937 (not prime) [0.2s]
Factors of F9: 2424833*552937374...<148 digits>...393118209 (last factor is not prime) [0.2s]
Factors of F10: 179769313...<309 digits>...224137217 (not prime) [0.2s]
Factors of F11: 319489*974849*103761886...<606 digits>...591348737 (last factor is not prime) [0.3s]
Factors of F12: 114689*910626896...<1,228 digits>...946770433 (last factor is not prime) [0.6s]
Factors of F13: 109074813...<2,467 digits>...715792897 (not prime) [1.3s]

PicoLisp[edit]

(seed (in "/dev/urandom" (rd 8)))
(de **Mod (X Y N)
   (let M 1
      (loop
         (when (bit? 1 Y)
            (setq M (% (* M X) N)) )
         (T (=0 (setq Y (>> 1 Y)))
            M )
         (setq X (% (* X X) N)) ) ) )
(de isprime (N)
   (cache '(NIL) N
      (if (== N 2)
         T
         (and
            (> N 1)
            (bit? 1 N)
            (let (Q (dec N)  N1 (dec N)  K 0  X)
               (until (bit? 1 Q)
                  (setq
                     Q (>> 1 Q)
                     K (inc K) ) )
               (catch 'composite
                  (do 16
                     (loop
                        (setq X
                           (**Mod
                              (rand 2 (min (dec N) 1000000000000))
                              Q
                              N ) )
                        (T (or (=1 X) (= X N1)))
                        (T
                           (do K
                              (setq X (**Mod X 2 N))
                              (when (=1 X) (throw 'composite))
                              (T (= X N1) T) ) )
                        (throw 'composite) ) )
                  (throw 'composite T) ) ) ) ) ) )
(de gcd (A B)
   (until (=0 B)
      (let M (% A B)
         (setq A B B M) ) )
   (abs A) )
(de g (A)
   (% (+ (% (* A A) N) C) N) )
(de pollard-brent (N)
   (let
      (A (dec N)
         Y (rand 1 (min A 1000000000000000000))
         C (rand 1 (min A 1000000000000000000))
         M (rand 1 (min A 1000000000000000000))
         G 1
         R 1
         Q 1 )
      (ifn (bit? 1 N)
         2
         (loop
            (NIL (=1 G))
            (setq X Y)
            (do R
               (setq Y (g Y)) )
            (zero K)
            (loop
               (NIL (and (> R K) (=1 G)))
               (setq YS Y)
               (do (min M (- R K))
                  (setq
                     Y (g Y)
                     Q (% (* Q (abs (- X Y))) N) ) )
               (setq
                  G (gcd Q N)
                  K (+ K M) )
            )
            (setq R (* R 2)) )
         (when (== G N)
            (loop
               (NIL (> G 1))
               (setq
                  YS (g YS)
                  G (gcd (abs (- X YS)) N) ) ) )
         (if (== G N)
            NIL
            G ) ) ) )
(de factors (N)
   (sort
      (make
         (loop
            (setq N (/ N (link (pollard-brent N))))
            (T (isprime N)) )
         (link N) ) ) )
(de fermat (N)
   (inc (** 2 (** 2 N))) )
(for (N 0 (>= 8 N) (inc N))
   (println N ': (fermat N)) )
(prinl)
(for (N 0 (>= 8 N) (inc N))
   (let N (fermat N)
      (println
         N
         ':
         (if (isprime N) 'PRIME (factors N)) ) ) )
Output:
$ pil VU.l 
0 : 3
1 : 5
2 : 17
3 : 257
4 : 65537
5 : 4294967297
6 : 18446744073709551617
7 : 340282366920938463463374607431768211457
8 : 115792089237316195423570985008687907853269984665640564039457584007913129639937

3 : PRIME
5 : PRIME
17 : PRIME
257 : PRIME
65537 : PRIME
4294967297 : (641 6700417)
18446744073709551617 : (274177 67280421310721)
340282366920938463463374607431768211457 : (59649589127497217 5704689200685129054721)
115792089237316195423570985008687907853269984665640564039457584007913129639937 : (1238926361552897 93461639715357977769163558199606896584051237541638188580280321)

Python[edit]

Procedural[edit]

def factors(x):
    factors = []
    i = 2
    s = int(x ** 0.5)
    while i < s:
        if x % i == 0:
            factors.append(i)
            x = int(x / i)
            s = int(x ** 0.5)
        i += 1
    factors.append(x)
    return factors

print("First 10 Fermat numbers:")
for i in range(10):
    fermat = 2 ** 2 ** i + 1
    print("F{} = {}".format(chr(i + 0x2080) , fermat))

print("\nFactors of first few Fermat numbers:")
for i in range(10):
    fermat = 2 ** 2 ** i + 1
    fac = factors(fermat)
    if len(fac) == 1:
        print("F{} -> IS PRIME".format(chr(i + 0x2080)))
    else:
        print("F{} -> FACTORS: {}".format(chr(i + 0x2080), fac))
Output:
First 10 Fermat numbers:
F₀ = 3
F₁ = 5
F₂ = 17
F₃ = 257
F₄ = 65537
F₅ = 4294967297
F₆ = 18446744073709551617
F₇ = 340282366920938463463374607431768211457
F₈ = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F₉ = 1340780792994259709957402499820584612747936582059239337772356144372176403007354697680187429816690342769003185818648605085375388281194656994643364
9006084097

Factors of first few Fermat numbers:
F₀ IS PRIME
F₁ IS PRIME
F₂ IS PRIME
F₃ IS PRIME
F₄ IS PRIME
F₅ FACTORS: [641, 6700417]
F₆ FACTORS: [274177, 67280421310721]
F₇ FACTORS: [59649589127497217, 5704689200685129054721]
F₈ FACTORS: [1238926361552897, 93461639715357977769163558199606896584051237541638188580280321]

Functional[edit]

Works with: Python version 3.7
'''Fermat numbers'''

from itertools import count, islice
from math import floor, sqrt


# fermat :: Int -> Int
def fermat(n):
    '''Nth Fermat number.
       Nth term of OEIS A000215.
    '''
    return 1 + (2 ** (2 ** n))


# fermats :: () -> [Int]
def fermats():
    '''Non-finite series of Fermat numbers.
       OEIS A000215.
    '''
    return (fermat(x) for x in enumFrom(0))


# --------------------------TEST---------------------------
# main :: IO ()
def main():
    '''First 10 Fermats, and factors of first 7.'''

    print(
        fTable('First ten Fermat numbers:')(str)(str)(
            fermat
        )(enumFromTo(0)(9))
    )

    print(
        fTable('\n\nFactors of first seven:')(str)(
            lambda xs: repr(xs) if 1 < len(xs) else '(prime)'
        )(primeFactors)(
            take(7)(fermats())
        )
    )


# -------------------------DISPLAY-------------------------

# fTable :: String -> (a -> String) ->
# (b -> String) -> (a -> b) -> [a] -> String
def fTable(s):
    '''Heading -> x display function -> fx display function ->
       f -> xs -> tabular string.
    '''
    def go(xShow, fxShow, f, xs):
        ys = [xShow(x) for x in xs]
        w = max(map(len, ys))
        return s + '\n' + '\n'.join(map(
            lambda x, y: y.rjust(w, ' ') + ' -> ' + fxShow(f(x)),
            xs, ys
        ))
    return lambda xShow: lambda fxShow: lambda f: lambda xs: go(
        xShow, fxShow, f, xs
    )


# -------------------------GENERIC-------------------------

# enumFrom :: Enum a => a -> [a]
def enumFrom(x):
    '''A non-finite stream of enumerable values,
       starting from the given value.
    '''
    return count(x) if isinstance(x, int) else (
        map(chr, count(ord(x)))
    )


# enumFromTo :: Int -> Int -> [Int]
def enumFromTo(m):
    '''Enumeration of integer values [m..n]'''
    def go(n):
        return list(range(m, 1 + n))
    return lambda n: go(n)


# primeFactors :: Int -> [Int]
def primeFactors(n):
    '''A list of the prime factors of n.
    '''
    def f(qr):
        r = qr[1]
        return step(r), 1 + r

    def step(x):
        return 1 + (x << 2) - ((x >> 1) << 1)

    def go(x):
        root = floor(sqrt(x))

        def p(qr):
            q = qr[0]
            return root < q or 0 == (x % q)

        q = until(p)(f)(
            (2 if 0 == x % 2 else 3, 1)
        )[0]
        return [x] if q > root else [q] + go(x // q)

    return go(n)


# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
    '''The prefix of xs of length n,
       or xs itself if n > length xs.
    '''
    return lambda xs: (
        xs[0:n]
        if isinstance(xs, (list, tuple))
        else list(islice(xs, n))
    )


# until :: (a -> Bool) -> (a -> a) -> a -> a
def until(p):
    '''The result of repeatedly applying f until p holds.
       The initial seed value is x.
    '''
    def go(f, x):
        v = x
        while not p(v):
            v = f(v)
        return v
    return lambda f: lambda x: go(f, x)


# MAIN ---
if __name__ == '__main__':
    main()
Output:
First ten Fermat numbers:
0 -> 3
1 -> 5
2 -> 17
3 -> 257
4 -> 65537
5 -> 4294967297
6 -> 18446744073709551617
7 -> 340282366920938463463374607431768211457
8 -> 115792089237316195423570985008687907853269984665640564039457584007913129639937
9 -> 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors of first seven:
                   3 -> (prime)
                   5 -> (prime)
                  17 -> (prime)
                 257 -> (prime)
               65537 -> (prime)
          4294967297 -> [641, 6700417]
18446744073709551617 -> [274177, 67280421310721]

Raku[edit]

(formerly Perl 6)

I gave up on factoring F₉ after about 20 minutes.

Library: ntheory
use ntheory:from<Perl5> <factor>;

my @Fermats = (^Inf).map: 2 ** 2 ** * + 1;

my $sub = '₀';
say "First 10 Fermat numbers:";
printf "F%s = %s\n", $sub++, $_ for @Fermats[^10];

$sub = '₀';
say "\nFactors of first few Fermat numbers:";
for @Fermats[^9].map( {"$_".&factor} ) -> $f {
    printf "Factors of F%s: %s %s\n", $sub++, $f.join(' '), $f.elems == 1 ?? '- prime' !! ''
}
Output:
First 10 Fermat numbers:
F₀ = 3
F₁ = 5
F₂ = 17
F₃ = 257
F₄ = 65537
F₅ = 4294967297
F₆ = 18446744073709551617
F₇ = 340282366920938463463374607431768211457
F₈ = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F₉ = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors of first few Fermat numbers:
Factors of F₀: 3 - prime
Factors of F₁: 5 - prime
Factors of F₂: 17 - prime
Factors of F₃: 257 - prime
Factors of F₄: 65537 - prime
Factors of F₅: 641 6700417 
Factors of F₆: 274177 67280421310721 
Factors of F₇: 59649589127497217 5704689200685129054721 
Factors of F₈: 1238926361552897 93461639715357977769163558199606896584051237541638188580280321

REXX[edit]

factoring by trial division[edit]

/*REXX program to find and display  Fermat  numbers, and show factors of Fermat numbers.*/
parse arg n .                                    /*obtain optional argument from the CL.*/
if n=='' | n==","  then n= 9                     /*Not specified?  Then use the default.*/
numeric digits 20                                /*ensure enough decimal digits, for n=9*/

       do j=0  to n;   f= 2** (2**j)   +  1      /*calculate a series of Fermat numbers.*/
       say right('F'j, length(n) + 1)': '     f  /*display a particular     "      "    */
       end   /*j*/
say
       do k=0  to n;   f= 2** (2**k)   +  1; say /*calculate a series of Fermat numbers.*/
       say center(' F'k": " f' ', 79, "═")       /*display a particular     "      "    */
       p= factr(f)                               /*factor a Fermat number,  given time. */
       if words(p)==1  then say f ' is prime.'
                       else say 'factors: '   p
       end   /*k*/
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
factr:  procedure; parse arg x 1 z,,?
             do k=1  to 11  by 2;  j= k;  if j==1  then j= 2;  if j==9  then iterate
             call build                          /*add  J  to the factors list.         */
             end   /*k*/                         /* [↑]  factor  X  with some low primes*/

             do y=0  by 2;         j= j + 2 +   y // 4      /*ensure not  ÷  by three.  */
             parse var j '' -1 _;  if _==5  then iterate    /*last digit a "5"? Skip it.*/
             if j*j>x | j>z  then leave
             call build                          /*add  Y  to the factors list.         */
             end   /*y*/                         /* [↑]  factor  X  with other higher #s*/
        j= z
        if z\==1  then ?= build()
        if ?=''   then do;  @.1= x;  ?= x;  #= 1;  end
        return ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
build:     do  while z//j==0;    z= z % j;    ?= ? j;    end;              return strip(?)
output   when using the default input:
F0:  3
F1:  5
F2:  17
F3:  257
F4:  65537
F5:  4294967297
F6:  18446744073709551617
F7:  340282366920938463463374607431768211457
F8:  115792089237316195423570985008687907853269984665640564039457584007913129639937
F9:  13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097


═══════════════════════════════════ F0:  3 ════════════════════════════════════
3  is prime.

═══════════════════════════════════ F1:  5 ════════════════════════════════════
5  is prime.

═══════════════════════════════════ F2:  17 ═══════════════════════════════════
17  is prime.

══════════════════════════════════ F3:  257 ═══════════════════════════════════
257  is prime.

═════════════════════════════════ F4:  65537 ══════════════════════════════════
65537  is prime.

═══════════════════════════════ F5:  4294967297 ═══════════════════════════════
factors:  641 6700417

══════════════════════════ F6:  18446744073709551617 ══════════════════════════
   ■  ■  ■   (the REXX program stopped via Ctrl─Alt─Break)   ■  ■  ■

factoring via Pollard's rho algorithm[edit]

/*REXX program to find and display  Fermat  numbers, and show factors of Fermat numbers.*/
parse arg n .                                    /*obtain optional argument from the CL.*/
if n=='' | n==","  then n= 9                     /*Not specified?  Then use the default.*/
numeric digits 200                               /*ensure enough decimal digits, for n=9*/

       do j=0  to n;   f= 2** (2**j)   +  1      /*calculate a series of Fermat numbers.*/
       say right('F'j, length(n) + 1)': '     f  /*display a particular     "      "    */
       end   /*j*/
say
       do k=5  to n;   f= 2** (2**k)   +  1; say /*calculate a series of Fermat numbers.*/
       say center(' F'k": " f' ', 79, "═")       /*display a particular     "      "    */
       a= rho(f)                                 /*factor a Fermat number,  given time. */
       b= f % a
       if a==b  then say f ' is prime.'
                else say 'factors:  '   commas(a)     " "     commas(b)
       end   /*k*/
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas:  parse arg _;  do ?=length(_)-3  to 1  by -3; _=insert(',', _, ?); end;   return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
rho:  procedure;  parse arg n;    y= 2;  d= 1    /*initialize  X, Y,  and  D  variables.*/
        do x=2  until d==n                       /*try rho method with X=2 for 1st time.*/
          do    while d==1
          x= (x*x + 1) // n
          v= (y*y + 1) // n
          y= (v*v + 1) // n
          parse value  x-y   with  xy  1  sig  2 /*obtain sign of the  x-y  difference. */
          if sig=='-'  then parse var  xy  2  xy /*Negative?   Then use absolute value. */
          nn= n
                do  until nn==0
                parse value xy//nn nn with nn xy /*assign two variables:   NN  and  XY  */
                end   /*until*/                  /*this is an  in-line   GCD  function. */
          d= xy                                  /*assign variable   D   with a new  XY */
          end   /*while*/
        end     /*x*/
      return d                                   /*found a factor of  N.      Return it.*/
output   when using the default input:
F0:  3
F1:  5
F2:  17
F3:  257
F4:  65537
F5:  4294967297
F6:  18446744073709551617
F7:  340282366920938463463374607431768211457
F8:  115792089237316195423570985008687907853269984665640564039457584007913129639937
F9:  13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097


═══════════════════════════════ F5:  4294967297 ═══════════════════════════════
factors:   641   6,700,417

══════════════════════════ F6:  18446744073709551617 ══════════════════════════
factors:   274,177   67,280,421,310,721

════════════════ F7:  340282366920938463463374607431768211457 ═════════════════
   ■  ■  ■   (the REXX program stopped via Ctrl─Alt─Break)   ■  ■  ■

Ring[edit]

decimals(0)
load "stdlib.ring"

see "working..." + nl
see "The first 10 Fermat numbers are:" + nl

num = 0
limit = 9

for n = 0 to limit 
    fermat = pow(2,pow(2,n)) + 1
    mod = fermat%2
    if n > 5
       ferm = string(fermat)
       tmp = number(right(ferm,1))+1
       fermat = left(ferm,len(ferm)-1) + string(tmp)
    ok
    see "F(" + n + ") = " + fermat + nl
next

see "done..." + nl

Output:

working...
The first 10 Fermat numbers are:
F(0) = 3
F(1) = 5
F(2) = 17
F(3) = 257
F(4) = 65537
F(5) = 4294967297
F(6) = 18446744073709551617
F(7) = 340282366920938463463374607431768211457
F(8) = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F(9) = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Ruby[edit]

This uses the `factor` function from the `coreutils` library
that comes standard with most GNU/Linux, BSD, and Unix systems.
https://www.gnu.org/software/coreutils/
https://en.wikipedia.org/wiki/GNU_Core_Utilities
def factors(n)
    factors = `factor #{n}`.split(' ')[1..-1].map(&:to_i)
    factors.group_by { _1 }.map { |prime, exp| [prime, exp.size] }             # Ruby 2.7 or later
    #factors.group_by { |prime| prime }.map { |prime, exp| [prime, exp.size] } # for all versions
end

def fermat(n); (1 << (1 << n)) | 1 end

puts "Value for each Fermat Number F0 .. F9."
(0..9).each { |n| puts "F#{n} = #{fermat(n)}" }
puts
puts "Factors for each Fermat Number F0 .. F8."
(0..8).each { |n| puts "F#{n} = #{factors fermat(n)}" }
System: Lenovo V570 (2011), I5-2410M, 2.9 GHz, Ruby 2.7.1
Run as: $ time ruby fermat.rb
Output:
Value for each Fermat Number F0 .. F9.
F0 = 3
F1 = 5
F2 = 17
F3 = 257
F4 = 65537
F5 = 4294967297
F6 = 18446744073709551617
F7 = 340282366920938463463374607431768211457
F8 = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F9 = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors for each Fermat Number F0 .. F8.
F0 = [[3, 1]]
F1 = [[5, 1]]
F2 = [[17, 1]]
F3 = [[257, 1]]
F4 = [[65537, 1]]
F5 = [[641, 1], [6700417, 1]]
F6 = [[274177, 1], [67280421310721, 1]]
F7 = [[59649589127497217, 1], [5704689200685129054721, 1]]
F8 = [[1238926361552897, 1], [93461639715357977769163558199606896584051237541638188580280321, 1]]
ruby fermat.rb  175.26s user 0.01s system 99% cpu 2:55.27 total

Rust[edit]

 struct DivisorGen {
    curr: u64,
    last: u64,
}

impl Iterator for DivisorGen {
    type Item = u64;

    fn next(&mut self) -> Option<u64> {
        self.curr += 2u64;

        if self.curr < self.last{
            None
        } else {
            Some(self.curr)
        }
    }
}

fn divisor_gen(num : u64) -> DivisorGen {
    DivisorGen { curr: 0u64, last: (num / 2u64) + 1u64 }
}

fn is_prime(num : u64) -> bool{
    if num == 2 || num == 3 {
        return true;
    } else if num % 2 == 0 || num % 3 == 0 || num <= 1{
        return false;
    }else{
        for i in divisor_gen(num){
            if num % i == 0{
                return false;
            }
        }
    }
    return true;
}


fn main() {
    let fermat_closure = |i : u32| -> u64 {2u64.pow(2u32.pow(i + 1u32))};
    let mut f_numbers : Vec<u64> = Vec::new();
    
    println!("First 4 Fermat numbers:");
    for i in 0..4 {
        let f = fermat_closure(i) + 1u64;
        f_numbers.push(f);
        println!("F{}: {}", i, f);
    }
    
    println!("Factor of the first four numbers:");
    for f in f_numbers.iter(){
        let is_prime : bool = f % 4 == 1 && is_prime(*f);
        let not_or_not = if is_prime {" "} else {" not "};
        println!("{} is{}prime", f, not_or_not);
    }
}
Output:
First 4 Fermat numbers:
F0: 5
F1: 17
F2: 257
F3: 65537
Factor of the first four numbers:
5 is prime
17 is prime
257 is prime
65537 is prime

Alternative using rug crate[edit]

Based on the C++ code, which was based on the Java solution.

// [dependencies]
// rug = "1.9"

use rug::Integer;

fn fermat(n: u32) -> Integer {
    Integer::from(Integer::u_pow_u(2, 2u32.pow(n))) + 1
}

fn g(x: Integer, n: &Integer) -> Integer {
    (Integer::from(&x * &x) + 1) % n
}

fn pollard_rho(n: &Integer) -> Integer {
    use rug::Assign;

    let mut x = Integer::from(2);
    let mut y = Integer::from(2);
    let mut d = Integer::from(1);
    let mut z = Integer::from(1);
    let mut count = 0;
    loop {
        x = g(x, n);
        y = g(g(y, n), n);
        d.assign(&x - &y);
        d = d.abs();
        z *= &d;
        z %= n;
        count += 1;
        if count == 100 {
            d.assign(z.gcd_ref(n));
            if d != 1 {
                break;
            }
            z.assign(1);
            count = 0;
        }
    }
    if d == *n {
        return Integer::from(0);
    }
    d
}

fn get_prime_factors(n: &Integer) -> Vec<Integer> {
    use rug::integer::IsPrime;
    let mut factors = Vec::new();
    let mut m = Integer::from(n);
    loop {
        if m.is_probably_prime(25) != IsPrime::No {
            factors.push(m);
            break;
        }
        let f = pollard_rho(&m);
        if f == 0 {
            factors.push(m);
            break;
        }
        factors.push(Integer::from(&f));
        m = m / f;
    }
    factors
}

fn main() {
    for i in 0..10 {
        println!("F({}) = {}", i, fermat(i));
    }
    println!("\nPrime factors:");
    for i in 0..9 {
        let f = get_prime_factors(&fermat(i));
        print!("F({}): {}", i, f[0]);
        for j in 1..f.len() {
            print!(", {}", f[j]);
        }
        println!();
    }
}
Output:

Execution time is about 8.5 minutes on my system (macOS 10.15, 3.2 GHz Quad-Core Intel Core i5).

F(0) = 3
F(1) = 5
F(2) = 17
F(3) = 257
F(4) = 65537
F(5) = 4294967297
F(6) = 18446744073709551617
F(7) = 340282366920938463463374607431768211457
F(8) = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F(9) = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Prime factors:
F(0): 3
F(1): 5
F(2): 17
F(3): 257
F(4): 65537
F(5): 641, 6700417
F(6): 274177, 67280421310721
F(7): 59649589127497217, 5704689200685129054721
F(8): 1238926361552897, 93461639715357977769163558199606896584051237541638188580280321

Scala[edit]

Translation of: Kotlin
import scala.collection.mutable
import scala.collection.mutable.ListBuffer

object FermatNumbers {
  def main(args: Array[String]): Unit = {
    println("First 10 Fermat numbers:")
    for (i <- 0 to 9) {
      println(f"F[$i] = ${fermat(i)}")
    }
    println()
    println("First 12 Fermat numbers factored:")
    for (i <- 0 to 12) {
      println(f"F[$i] = ${getString(getFactors(i, fermat(i)))}")
    }
  }

  private val TWO = BigInt(2)

  def fermat(n: Int): BigInt = {
    TWO.pow(math.pow(2.0, n).intValue()) + 1
  }

  def getString(factors: List[BigInt]): String = {
    if (factors.size == 1) {
      return s"${factors.head} (PRIME)"
    }

    factors.map(a => a.toString)
      .map(a => if (a.startsWith("-")) "(C" + a.replace("-", "") + ")" else a)
      .reduce((a, b) => a + " * " + b)
  }

  val COMPOSITE: mutable.Map[Int, String] = scala.collection.mutable.Map(
    9 -> "5529",
    10 -> "6078",
    11 -> "1037",
    12 -> "5488",
    13 -> "2884"
  )

  def getFactors(fermatIndex: Int, n: BigInt): List[BigInt] = {
    var n2 = n
    var factors = new ListBuffer[BigInt]
    var loop = true
    while (loop) {
      if (n2.isProbablePrime(100)) {
        factors += n2
        loop = false
      } else {
        if (COMPOSITE.contains(fermatIndex)) {
          val stop = COMPOSITE(fermatIndex)
          if (n2.toString.startsWith(stop)) {
            factors += -n2.toString().length
            loop = false
          }
        }
        if (loop) {
          val factor = pollardRhoFast(n2)
          if (factor == 0) {
            factors += n2
            loop = false
          } else {
            factors += factor
            n2 = n2 / factor
          }
        }
      }
    }

    factors.toList
  }

  def pollardRhoFast(n: BigInt): BigInt = {
    var x = BigInt(2)
    var y = BigInt(2)
    var z = BigInt(1)
    var d = BigInt(1)
    var count = 0

    var loop = true
    while (loop) {
      x = pollardRhoG(x, n)
      y = pollardRhoG(pollardRhoG(y, n), n)
      d = (x - y).abs
      z = (z * d) % n
      count += 1
      if (count == 100) {
        d = z.gcd(n)
        if (d != 1) {
          loop = false
        } else {
          z = BigInt(1)
          count = 0
        }
      }
    }

    println(s"    Pollard rho try factor $n")
    if (d == n) {
      return 0
    }
    d
  }

  def pollardRhoG(x: BigInt, n: BigInt): BigInt = ((x * x) + 1) % n
}
Output:
First 10 Fermat numbers:
F[0] = 3
F[1] = 5
F[2] = 17
F[3] = 257
F[4] = 65537
F[5] = 4294967297
F[6] = 18446744073709551617
F[7] = 340282366920938463463374607431768211457
F[8] = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F[9] = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

First 12 Fermat numbers factored:
F[0] = 3 (PRIME)
F[1] = 5 (PRIME)
F[2] = 17 (PRIME)
F[3] = 257 (PRIME)
F[4] = 65537 (PRIME)
    Pollard rho try factor 4294967297
F[5] = 641 * 6700417
    Pollard rho try factor 18446744073709551617
F[6] = 274177 * 67280421310721
    Pollard rho try factor 340282366920938463463374607431768211457
F[7] = 59649589127497217 * 5704689200685129054721
    Pollard rho try factor 115792089237316195423570985008687907853269984665640564039457584007913129639937
F[8] = 1238926361552897 * 93461639715357977769163558199606896584051237541638188580280321
    Pollard rho try factor 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097
F[9] = 2424833 * (C148)
    Pollard rho try factor 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137217
    Pollard rho try factor 3942951359960012586542991835686376608231592127249807732373409846031135195659174148737161255930050543559319182152642816343958573976075461198274610155058226350701077796608546283231637018483208223116080561800334422176622099740983337736621316898600121619871377542107047343253864459964167331555646795960321
F[10] = 45592577 * 6487031809 * (C291)
    Pollard rho try factor 32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638215525166389437335543602135433229604645318478604952148193555853611059596230657
    Pollard rho try factor 33150781373639412155846573868024639672856106606987835072026893834352453701925006737655987144186344206834820532125383540932102878651453631377873037143648178457002958783669056532601662155256508553423204658756451069116132055982639112479817996775373591674794399801442382402697828988429044712163168243619196804348072710121945390948428910309765481110260333687910970886853046635254307274981520537180895290310783635953818082306553996934497908037349010876970379631341148456045116407475229712217130141926525362871253437794629422541384355185626695660779797862427347553871011957167960991543632376506466281643163047416635393
F[11] = 974849 * 319489 * (C606)
    Pollard rho try factor 1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085903001302413467189726673216491511131602920781738033436090243804708340403154190337
    Pollard rho try factor 9106268965752186405773463110818163752233991481723476361152625650968740750826648212547208641935996986118024454955917854702609434541985662158212523327009262247869952450049350838706079834460006786304075107567909269645531121898331250125751682239313156601738683820643686003638396435055834553570682260579462973839574318172464558815116581626749391315641251152532705571615644886981829338611134458123396450764186936496833100701185274214915961723337127995182593580031119299575446791424418154036863609858251201843852076223383379133238000289598800458955855329052103961332983048473420515918928565951506637819342706575976725030506905683310915700945442329953941604008255667676914945655757474715779252371155778495946746587469464160684843488975918662295274965457887082037460184558511575570318625886351712499453155527762335682281851520733417380809781252979478377941937578568481859702438295520231435016188495646093490407803983345420364088331996467459309353537828143080691834120737157445502646809195267166779721413577366833939771467773331873590129210913628329073978766992198221682739812652450408607796042492802295258713711959073218748776359806123717024800451461326745599716651128725627280643537507664130920416107218492950792456907321580171946770433
    Pollard rho try factor 350001591824186871106763863899530746217943677302816436473017740242946077356624684213115564488348300185108877411543625345263121839042445381828217455916005721464151569276047005167043946981206545317048033534893189043572263100806868980998952610596646556521480658280419327021257968923645235918768446677220584153297488306270426504473941414890547838382804073832577020334339845312084285496895699728389585187497914849919557000902623608963141559444997044721763816786117073787751589515083702681348245404913906488680729999788351730419178916889637812821243344085799435845038164784900107242721493170135785069061880328434342030106354742821995535937481028077744396075726164309052460585559946407282864168038994640934638329525854255227752926198464207255983432778402344965903148839661825873175277621985527846249416909718758069997783882773355041329208102046913755441975327368023946523920699020098723785533557579080342841062805878477869513695185309048285123705067072486920463781103076554014502567884803571416673251784936825115787932810954867447447568320403976197134736485611912650805539603318790667901618038578533362100071745480995207732506742832634459994375828162163700807237997808869771569154136465922798310222055287047244647419069003284481
F[12] = 114689 * 26017793 * 63766529 * (C1213)

Sidef[edit]

func fermat_number(n) {
    2**(2**n) + 1
}

func fermat_one_factor(n) {
    fermat_number(n).ecm_factor
}

for n in (0..9) {
    say "F_#{n} = #{fermat_number(n)}"
}

say ''

for n in (0..13) {
    var f = fermat_one_factor(n)
    say ("F_#{n} = ", join(' * ', f.shift,
      f.map { <C P>[.is_prime] + .len }...))
}
Output:
F_0 = 3
F_1 = 5
F_2 = 17
F_3 = 257
F_4 = 65537
F_5 = 4294967297
F_6 = 18446744073709551617
F_7 = 340282366920938463463374607431768211457
F_8 = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F_9 = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

F_0 = 3
F_1 = 5
F_2 = 17
F_3 = 257
F_4 = 65537
F_5 = 641 * P7
F_6 = 274177 * P14
F_7 = 59649589127497217 * P22
F_8 = 1238926361552897 * P62
F_9 = 2424833 * C148
F_10 = 45592577 * C301
F_11 = 319489 * C612
F_12 = 114689 * C1228
F_13 = 2710954639361 * C2454

Tcl[edit]

namespace import ::tcl::mathop::*
package require math::numtheory 1.1.1; # Buggy before tcllib-1.20

proc fermat n {
	+ [** 2 [** 2 $n]] 1
}


for {set i 0} {$i < 10} {incr i} {
	puts "F$i = [fermat $i]"
}

for {set i 1} {1} {incr i} {
	puts -nonewline "F$i... "
	flush stdout
	set F [fermat $i]
	set factors [math::numtheory::primeFactors $F]
	if {[llength $factors] == 1} {
		puts "is prime"
	} else {
		puts "factors: $factors"
	}
}
Output:
$ tclsh fermat.tcl
F0 = 3
F1 = 5
F2 = 17
F3 = 257
F4 = 65537
F5 = 4294967297
F6 = 18446744073709551617
F7 = 340282366920938463463374607431768211457
F8 = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F9 = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097
F1... is prime
F2... is prime
F3... is prime
F4... is prime
F5... factors: 641 6700417
F6...

Wren[edit]

Wren-cli[edit]

Library: Wren-big

The first seven Fermat numbers are factorized in about 0.75 seconds but as it would take 'hours' to get any further I've stopped there.

import "/big" for BigInt

var fermat = Fn.new { |n| BigInt.two.pow(2.pow(n)) + 1 }

var fns = List.filled(10, null)
System.print("The first 10 Fermat numbers are:")
for (i in 0..9) {
    fns[i] = fermat.call(i)
    System.print("F%(String.fromCodePoint(0x2080+i)) = %(fns[i])")
}

System.print("\nFactors of the first 7 Fermat numbers:")
for (i in 0..6) {
    System.write("F%(String.fromCodePoint(0x2080+i)) = ")
    var factors = BigInt.primeFactors(fns[i])
    System.write("%(factors)")
    if (factors.count == 1) {
        System.print(" (prime)")
    } else if (!factors[1].isProbablePrime(5)) {
        System.print(" (second factor is composite)")
    } else {
        System.print()
    }
}
Output:
The first 10 Fermat numbers are:
F₀ = 3
F₁ = 5
F₂ = 17
F₃ = 257
F₄ = 65537
F₅ = 4294967297
F₆ = 18446744073709551617
F₇ = 340282366920938463463374607431768211457
F₈ = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F₉ = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors of the first 7 Fermat numbers:
F₀ = [3] (prime)
F₁ = [5] (prime)
F₂ = [17] (prime)
F₃ = [257] (prime)
F₄ = [65537] (prime)
F₅ = [641, 6700417]
F₆ = [274177, 67280421310721]

Embedded[edit]

Library: Wren-gmp
Library: Wren-ecm

This uses Lenstra's elliptical curve method (ECM) to find F₇ and Pollard's Rho to find the rest, including the first factor of F₉.

The overall time for this particular run was about 86 seconds with F₇ being found in only a couple of them. However, being non-deterministic, ECM is quite erratic time-wise and could take up to 3 minutes longer.

/* fermat_numbers_gmp.wren */

import "./gmp" for Mpz
import "./ecm" for Ecm
import "random" for Random
 
var fermat = Fn.new { |n| Mpz.two.pow(2.pow(n)) + 1 }
   
var fns = List.filled(10, null)
System.print("The first 10 Fermat numbers are:")
for (i in 0..9) {
    fns[i] = fermat.call(i)
    System.print("F%(String.fromCodePoint(0x2080+i)) = %(fns[i])")
}

System.print("\nFactors of the first 8 Fermat numbers:")
for (i in 0..8) {
    System.write("F%(String.fromCodePoint(0x2080+i)) = ")
    var factors = (i != 7) ? Mpz.primeFactors(fns[i]) : Ecm.primeFactors(fns[i])
    System.write("%(factors)")
    if (factors.count == 1) {
        System.print(" (prime)")
    } else if (!factors[1].probPrime(15)) {
        System.print(" (second factor is composite)")
    } else {
        System.print()
    }
}

System.print("\nThe first factor of F₉ is %(Mpz.pollardRho(fns[9])).")
Output:
The first 10 Fermat numbers are:
F₀ = 3
F₁ = 5
F₂ = 17
F₃ = 257
F₄ = 65537
F₅ = 4294967297
F₆ = 18446744073709551617
F₇ = 340282366920938463463374607431768211457
F₈ = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F₉ = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

Factors of the first 8 Fermat numbers:
F₀ = [3] (prime)
F₁ = [5] (prime)
F₂ = [17] (prime)
F₃ = [257] (prime)
F₄ = [65537] (prime)
F₅ = [641, 6700417]
F₆ = [274177, 67280421310721]
F₈ = [1238926361552897, 93461639715357977769163558199606896584051237541638188580280321]

The first factor of F₉ is 2424833.

zkl[edit]

Library: GMP
GNU Multiple Precision Arithmetic Library

for big ints and primes

fermatsW:=[0..].tweak(fcn(n){ BI(2).pow(BI(2).pow(n)) + 1 });
println("First 10 Fermat numbers:");
foreach n in (10){ println("F",n,": ",fermatsW.next()) }
Output:
First 10 Fermat numbers:
F0: 3
F1: 5
F2: 17
F3: 257
F4: 65537
F5: 4294967297
F6: 18446744073709551617
F7: 340282366920938463463374607431768211457
F8: 115792089237316195423570985008687907853269984665640564039457584007913129639937
F9: 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097
fcn primeFactorsBI(n){  // Return a list of the prime factors of n
   acc:=fcn(n,k,acc,maxD){  // k is primes
      if(n==1 or k>maxD) acc.close();
      else{
	 q,r:=n.div2(k);   // divr-->(quotient,remainder)
	 if(r==0) return(self.fcn(q,k,acc.write(k.copy()),q.root(2)));
	 return(self.fcn(n, k.nextPrime(), acc,maxD)) # both are tail recursion
      }
   }(n,BI(2),Sink(List),n.root(2));
   m:=acc.reduce('*,BI(1));  // mulitply factors
   if(n!=m) acc.append(n/m); // opps, missed last factor
   else acc;
}
fermatsW:=[0..].tweak(fcn(n){ BI(2).pow(BI(2).pow(n)) + 1 });
println("Factors of first few Fermat numbers:");
foreach n in (7){
   println("Factors of F",n,": ",factorsBI(fermatsW.next()).concat(" "));
}
Output:
Factors of first few Fermat numbers:
Factors of F0: 3
Factors of F1: 5
Factors of F2: 17
Factors of F3: 257
Factors of F4: 65537
Factors of F5: 641 6700417
Factors of F6: 274177 67280421310721