Multi-base primes
You are encouraged to solve this task according to the task description, using any language you may know.
Prime numbers are prime no matter what base they are represented in.
A prime number in a base other than 10 may not look prime at first glance.
For instance: 19 base 10 is 25 in base 7.
Several different prime numbers may be expressed as the "same" string when converted to a different base.
- 107 base 10 converted to base 6 == 255
- 173 base 10 converted to base 8 == 255
- 353 base 10 converted to base 12 == 255
- 467 base 10 converted to base 14 == 255
- 743 base 10 converted to base 18 == 255
- 1277 base 10 converted to base 24 == 255
- 1487 base 10 converted to base 26 == 255
- 2213 base 10 converted to base 32 == 255
- Task
Restricted to bases 2 through 36; find the strings that have the most different bases that evaluate to that string when converting prime numbers to a base.
Find the conversion string, the amount of bases that evaluate a prime to that string and the enumeration of bases that evaluate a prime to that string.
Display here, on this page, the string, the count and the list for all of the: 1 character, 2 character, 3 character, and 4 character strings that have the maximum base count that evaluate to that string.
Should be no surprise, the string '2' has the largest base count for single character strings.
- Stretch goal
Do the same for the maximum 5 character string.
C++
Originally translated from Wren with ideas borrowed from other solutions. The maximum base and number of characters can be specified as command line arguments. <lang cpp>#include <algorithm>
- include <cmath>
- include <cstdint>
- include <cstdlib>
- include <cstring>
- include <iostream>
- include <string>
- include <vector>
- include <primesieve.hpp>
class prime_sieve { public:
explicit prime_sieve(uint64_t limit); bool is_prime(uint64_t n) const { return n == 2 || ((n & 1) == 1 && sieve[n >> 1]); }
private:
std::vector<bool> sieve;
};
prime_sieve::prime_sieve(uint64_t limit) : sieve((limit + 1) / 2, false) {
primesieve::iterator iter; uint64_t prime = iter.next_prime(); // consume 2 while ((prime = iter.next_prime()) <= limit) { sieve[prime >> 1] = true; }
}
template <typename T> void print(std::ostream& out, const std::vector<T>& v) {
if (!v.empty()) { out << '['; auto i = v.begin(); out << *i++; for (; i != v.end(); ++i) out << ", " << *i; out << ']'; }
}
std::string to_string(const std::vector<unsigned int>& v) {
static constexpr char digits[] = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ"; std::string str; for (auto i : v) str += digits[i]; return str;
}
bool increment(std::vector<unsigned int>& digits, unsigned int max_base) {
for (auto i = digits.rbegin(); i != digits.rend(); ++i) { if (*i + 1 != max_base) { ++*i; return true; } *i = 0; } return false;
}
void multi_base_primes(unsigned int max_base, unsigned int max_length) {
prime_sieve sieve(static_cast<uint64_t>(std::pow(max_base, max_length))); for (unsigned int length = 1; length <= max_length; ++length) { std::cout << length << "-character strings which are prime in most bases: "; unsigned int most_bases = 0; std::vector< std::pair<std::vector<unsigned int>, std::vector<unsigned int>>> max_strings; std::vector<unsigned int> digits(length, 0); digits[0] = 1; std::vector<unsigned int> bases; do { auto max = std::max_element(digits.begin(), digits.end()); unsigned int min_base = 2; if (max != digits.end()) min_base = std::max(min_base, *max + 1); if (most_bases > max_base - min_base + 1) continue; bases.clear(); for (unsigned int b = min_base; b <= max_base; ++b) { if (max_base - b + 1 + bases.size() < most_bases) break; uint64_t n = 0; for (auto d : digits) n = n * b + d; if (sieve.is_prime(n)) bases.push_back(b); } if (bases.size() > most_bases) { most_bases = bases.size(); max_strings.clear(); } if (bases.size() == most_bases) max_strings.emplace_back(digits, bases); } while (increment(digits, max_base)); std::cout << most_bases << '\n'; for (const auto& m : max_strings) { std::cout << to_string(m.first) << " -> "; print(std::cout, m.second); std::cout << '\n'; } std::cout << '\n'; }
}
int main(int argc, char** argv) {
unsigned int max_base = 36; unsigned int max_length = 4; for (int arg = 1; arg + 1 < argc; ++arg) { if (strcmp(argv[arg], "-max_base") == 0) max_base = strtoul(argv[++arg], nullptr, 10); else if (strcmp(argv[arg], "-max_length") == 0) max_length = strtoul(argv[++arg], nullptr, 10); } if (max_base > 62) { std::cerr << "Max base cannot be greater than 62.\n"; return EXIT_FAILURE; } if (max_base < 2) { std::cerr << "Max base cannot be less than 2.\n"; return EXIT_FAILURE; } multi_base_primes(max_base, max_length); return EXIT_SUCCESS;
}</lang>
- Output:
Maximum base 36 and maximum length 6. This takes 0.41 seconds to process up to 5 character strings and 15 seconds to process up to 6 characters (3.2GHz Intel Core i5-4570).
1-character strings which are prime in most bases: 34 2 -> [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] 2-character strings which are prime in most bases: 18 21 -> [3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36] 3-character strings which are prime in most bases: 18 131 -> [4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 19, 20, 23, 25, 27, 29, 30, 34] 551 -> [6, 7, 11, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 30, 32, 35, 36] 737 -> [8, 9, 11, 12, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 29, 30, 31, 36] 4-character strings which are prime in most bases: 19 1727 -> [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36] 5347 -> [8, 9, 10, 11, 12, 13, 16, 18, 19, 22, 24, 25, 26, 30, 31, 32, 33, 34, 36] 5-character strings which are prime in most bases: 18 30271 -> [8, 10, 12, 13, 16, 17, 18, 20, 21, 23, 24, 25, 31, 32, 33, 34, 35, 36] 6-character strings which are prime in most bases: 18 441431 -> [5, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 23, 26, 28, 30, 31, 32, 33]
Maximum base 62 and maximum length 5. This takes 0.15 seconds to process up to 4 character strings and 6.4 seconds to process up to 5 characters (3.2GHz Intel Core i5-4570).
1-character strings which are prime in most bases: 60 2 -> [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] 2-character strings which are prime in most bases: 31 65 -> [7, 8, 9, 11, 13, 14, 16, 17, 18, 21, 22, 24, 27, 28, 29, 31, 32, 37, 38, 39, 41, 42, 43, 44, 46, 48, 51, 52, 57, 58, 59] 3-character strings which are prime in most bases: 33 1l1 -> [22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62] b9b -> [13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 30, 31, 34, 36, 39, 40, 42, 45, 47, 49, 50, 52, 53, 54, 57, 58, 59, 60, 61, 62] 4-character strings which are prime in most bases: 32 1727 -> [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36, 37, 38, 39, 41, 45, 46, 48, 50, 51, 57, 58, 60, 61] 417b -> [12, 13, 15, 16, 17, 18, 19, 21, 23, 25, 28, 30, 32, 34, 35, 37, 38, 39, 41, 45, 48, 49, 50, 51, 52, 54, 56, 57, 58, 59, 61, 62] 5-character strings which are prime in most bases: 30 50161 -> [7, 8, 9, 13, 17, 18, 19, 20, 25, 28, 29, 30, 31, 33, 35, 36, 38, 39, 41, 42, 43, 44, 47, 48, 52, 55, 56, 59, 60, 62]
F#
This task uses Extensible Prime Generator (F#) <lang fsharp> // Multi-base primes. Nigel Galloway: July 4th., 2021 let digits="0123456789abcdefghijklmnopqrstuvwxyz" let fG n g=let rec fN g=function i when i<n->i::g |i->fN((i%n)::g)(i/n) in primes32()|>Seq.skipWhile((>)(pown n (g-1)))|>Seq.takeWhile((>)(pown n g))|>Seq.map(fun g->(n,fN [] g)) let fN g={2..36}|>Seq.collect(fun n->fG n g)|>Seq.groupBy snd|>Seq.groupBy(snd>>(Seq.length))|>Seq.maxBy fst {1..4}|>Seq.iter(fun g->let n,i=fN g in printfn "The following strings of length %d represent primes in the maximum number of bases(%d):" g n
i|>Seq.iter(fun(n,g)->printf " %s->" (n|>List.map(fun n->digits.[n])|>Array.ofList|>System.String) g|>Seq.iter(fun(g,_)->printf "%d " g); printfn ""); printfn "")
</lang>
- Output:
The following strings of length 1 represent primes in the maximum number of bases(34): 2->3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 The following strings of length 2 represent primes in the maximum number of bases(18): 21->3 5 6 8 9 11 14 15 18 20 21 23 26 29 30 33 35 36 The following strings of length 3 represent primes in the maximum number of bases(18): 131->4 5 7 8 9 10 12 14 15 18 19 20 23 25 27 29 30 34 551->6 7 11 13 14 15 16 17 19 21 22 24 25 26 30 32 35 36 737->8 9 11 12 13 15 16 17 19 22 23 24 25 26 29 30 31 36 The following strings of length 4 represent primes in the maximum number of bases(19): 1727->8 9 11 12 13 15 16 17 19 20 22 23 24 26 27 29 31 33 36 5347->8 9 10 11 12 13 16 18 19 22 24 25 26 30 31 32 33 34 36
Factor
<lang factor>USING: assocs assocs.extras formatting io kernel math math.functions math.parser math.primes math.ranges present sequences ;
- prime?* ( n -- ? ) [ prime? ] [ f ] if* ; inline
- (bases) ( n -- range quot )
present 2 36 [a,b] [ base> prime?* ] with ; inline
- <digits> ( n -- range ) [ 1 - ] keep [ 10^ ] bi@ [a,b) ;
- multibase ( n -- assoc )
<digits> [ (bases) count ] zip-with assoc-invert expand-keys-push-at >alist [ first ] supremum-by ;
- multibase. ( n -- )
dup multibase first2 [ "%d-digit numbers that are prime in the most bases: %d\n" printf ] dip [ dup (bases) filter "%d => %[%d, %]\n" printf ] each ;
4 [1,b] [ multibase. nl ] each</lang>
- Output:
1-digit numbers that are prime in the most bases: 34 2 => { 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 } 2-digit numbers that are prime in the most bases: 18 21 => { 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36 } 3-digit numbers that are prime in the most bases: 18 131 => { 4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 19, 20, 23, 25, 27, 29, 30, 34 } 551 => { 6, 7, 11, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 30, 32, 35, 36 } 737 => { 8, 9, 11, 12, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 29, 30, 31, 36 } 4-digit numbers that are prime in the most bases: 19 1727 => { 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36 } 5347 => { 8, 9, 10, 11, 12, 13, 16, 18, 19, 22, 24, 25, 26, 30, 31, 32, 33, 34, 36 }
Go
This takes about 1.2 seconds and 31.3 seconds to process up to 5 and 6 character strings, respectively. <lang go>package main
import (
"fmt" "math" "rcu"
)
var maxDepth = 6 var maxBase = 36 var c = rcu.PrimeSieve(int(math.Pow(float64(maxBase), float64(maxDepth))), true) var digits = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ" var maxStrings [][][]int var mostBases = -1
func maxSlice(a []int) int {
max := 0 for _, e := range a { if e > max { max = e } } return max
}
func maxInt(a, b int) int {
if a > b { return a } return b
}
func process(indices []int) {
minBase := maxInt(2, maxSlice(indices)+1) if maxBase - minBase + 1 < mostBases { return // can't affect results so return } var bases []int for b := minBase; b <= maxBase; b++ { n := 0 for _, i := range indices { n = n*b + i } if !c[n] { bases = append(bases, b) } } count := len(bases) if count > mostBases { mostBases = count indices2 := make([]int, len(indices)) copy(indices2, indices) maxStrings = [][][]int{[][]int{indices2, bases}} } else if count == mostBases { indices2 := make([]int, len(indices)) copy(indices2, indices) maxStrings = append(maxStrings, [][]int{indices2, bases}) }
}
func printResults() {
fmt.Printf("%d\n", len(maxStrings[0][1])) for _, m := range maxStrings { s := "" for _, i := range m[0] { s = s + string(digits[i]) } fmt.Printf("%s -> %v\n", s, m[1]) }
}
func nestedFor(indices []int, length, level int) {
if level == len(indices) { process(indices) } else { indices[level] = 0 if level == 0 { indices[level] = 1 } for indices[level] < length { nestedFor(indices, length, level+1) indices[level]++ } }
}
func main() {
for depth := 1; depth <= maxDepth; depth++ { fmt.Print(depth, " character strings which are prime in most bases: ") maxStrings = maxStrings[:0] mostBases = -1 indices := make([]int, depth) nestedFor(indices, maxBase, 0) printResults() fmt.Println() }
}</lang>
- Output:
1 character strings which are prime in most bases: 34 2 -> [3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36] 2 character strings which are prime in most bases: 18 21 -> [3 5 6 8 9 11 14 15 18 20 21 23 26 29 30 33 35 36] 3 character strings which are prime in most bases: 18 131 -> [4 5 7 8 9 10 12 14 15 18 19 20 23 25 27 29 30 34] 551 -> [6 7 11 13 14 15 16 17 19 21 22 24 25 26 30 32 35 36] 737 -> [8 9 11 12 13 15 16 17 19 22 23 24 25 26 29 30 31 36] 4 character strings which are prime in most bases: 19 1727 -> [8 9 11 12 13 15 16 17 19 20 22 23 24 26 27 29 31 33 36] 5347 -> [8 9 10 11 12 13 16 18 19 22 24 25 26 30 31 32 33 34 36] 5 character strings which are prime in most bases: 18 30271 -> [8 10 12 13 16 17 18 20 21 23 24 25 31 32 33 34 35 36] 6 character strings which are prime in most bases: 18 441431 -> [5 8 9 11 12 14 16 17 19 21 22 23 26 28 30 31 32 33]
If we change maxBase to 62 and maxDepth to 5 in the above code, then the following results are reached in 0.5 and 19.2 seconds for 4 and 5 digit character strings, respectively:
1 character strings which are prime in most bases: 60 2 -> [3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62] 2 character strings which are prime in most bases: 31 65 -> [7 8 9 11 13 14 16 17 18 21 22 24 27 28 29 31 32 37 38 39 41 42 43 44 46 48 51 52 57 58 59] 3 character strings which are prime in most bases: 33 1l1 -> [22 23 25 26 27 28 29 30 31 32 33 34 36 38 39 40 41 42 43 44 45 46 48 51 52 53 54 57 58 59 60 61 62] b9b -> [13 14 15 16 17 19 20 21 23 24 26 27 28 30 31 34 36 39 40 42 45 47 49 50 52 53 54 57 58 59 60 61 62] 4 character strings which are prime in most bases: 32 1727 -> [8 9 11 12 13 15 16 17 19 20 22 23 24 26 27 29 31 33 36 37 38 39 41 45 46 48 50 51 57 58 60 61] 417b -> [12 13 15 16 17 18 19 21 23 25 28 30 32 34 35 37 38 39 41 45 48 49 50 51 52 54 56 57 58 59 61 62] 5 character strings which are prime in most bases: 30 50161 -> [7 8 9 13 17 18 19 20 25 28 29 30 31 33 35 36 38 39 41 42 43 44 47 48 52 55 56 59 60 62]
Julia
<lang julia>using Primes
function maxprimebases(ndig, maxbase)
maxprimebases = [Int[]] nwithbases = [0] maxprime = 10^(ndig) - 1 for p in div(maxprime + 1, 10):maxprime dig = digits(p) bases = [b for b in 2:maxbase if (isprime(evalpoly(b, dig)) && all(i -> i < b, dig))] if length(bases) > length(first(maxprimebases)) maxprimebases = [bases] nwithbases = [p] elseif length(bases) == length(first(maxprimebases)) push!(maxprimebases, bases) push!(nwithbases, p) end end alen, vlen = length(first(maxprimebases)), length(maxprimebases) println("\nThe maximum number of prime valued bases for base 10 numeric strings of length ", ndig, " is $alen. The base 10 value list of ", vlen > 1 ? "these" : "this", " is:") for i in eachindex(maxprimebases) println(nwithbases[i], " => ", maxprimebases[i]) end
end
@time for n in 1:6
maxprimebases(n, 36)
end
</lang>
- Output:
The maximum number of prime valued bases for base 10 numeric strings of length 1 is 34. The base 10 value list of this is: 2 => [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] The maximum number of prime valued bases for base 10 numeric strings of length 2 is 18. The base 10 value list of this is: 21 => [3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36] The maximum number of prime valued bases for base 10 numeric strings of length 3 is 18. The base 10 value list of these is: 131 => [4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 19, 20, 23, 25, 27, 29, 30, 34] 551 => [6, 7, 11, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 30, 32, 35, 36] 737 => [8, 9, 11, 12, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 29, 30, 31, 36] The maximum number of prime valued bases for base 10 numeric strings of length 4 is 19. The base 10 value list of these is: 1727 => [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36] 5347 => [8, 9, 10, 11, 12, 13, 16, 18, 19, 22, 24, 25, 26, 30, 31, 32, 33, 34, 36] The maximum number of prime valued bases for base 10 numeric strings of length 5 is 18. The base 10 value list of this is: 30271 => [8, 10, 12, 13, 16, 17, 18, 20, 21, 23, 24, 25, 31, 32, 33, 34, 35, 36] The maximum number of prime valued bases for base 10 numeric strings of length 6 is 18. The base 10 value list of this is: 441431 => [5, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 23, 26, 28, 30, 31, 32, 33] 4.808196 seconds (8.58 M allocations: 357.983 MiB, 0.75% gc time)
Up to base 62
<lang julia>using Primes
function maxprimebases(ndig, maxbase)
maxprimebases = [Int[]] nwithbases = ["0"] for tup in Iterators.product([0:maxbase-1 for _ in 1:ndig - 1]..., 1:maxbase-1) dig = collect(tup) foundbases = Int[] for b in maximum(dig)+1:maxbase if isprime(evalpoly(b, dig)) push!(foundbases, b) end maxbase - b + length(foundbases) < length(maxprimebases) && break # shortcut if hopeless end if length(foundbases) > length(first(maxprimebases)) maxprimebases = [foundbases] nwithbases = [prod(string.(reverse(dig), base = any(x -> x > 9, dig) ? 32 : 10))] elseif length(foundbases) == length(first(maxprimebases)) push!(maxprimebases, foundbases) push!(nwithbases, prod(string.(reverse(dig), base = any(x -> x > 9, dig) ? 32 : 10))) end end alen, vlen = length(first(maxprimebases)), length(maxprimebases) println("\nThe maximum number of prime valued bases for base up to $maxbase numeric strings of length ", ndig, " is $alen. The value list of ", vlen > 1 ? "these" : "this", " is:") for i in eachindex(maxprimebases) println(nwithbases[i], maxprimebases[i][1] > 10 ? "(base 32)" : "", " => ", maxprimebases[i]) end
end
for n in 1:5
maxprimebases(n, 36) maxprimebases(n, 62)
end
</lang>
- Output:
The maximum number of prime valued bases for base up to 36 numeric strings of length 1 is 34. The value list of this is: 2 => [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] The maximum number of prime valued bases for base up to 62 numeric strings of length 1 is 60. The value list of this is: 2 => [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] The maximum number of prime valued bases for base up to 36 numeric strings of length 2 is 18. The value list of this is: 21 => [3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36] The maximum number of prime valued bases for base up to 62 numeric strings of length 2 is 31. The value list of this is: 65 => [7, 8, 9, 11, 13, 14, 16, 17, 18, 21, 22, 24, 27, 28, 29, 31, 32, 37, 38, 39, 41, 42, 43, 44, 46, 48, 51, 52, 57, 58, 59] The maximum number of prime valued bases for base up to 36 numeric strings of length 3 is 18. The value list of these is: 131 => [4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 19, 20, 23, 25, 27, 29, 30, 34] 551 => [6, 7, 11, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 30, 32, 35, 36] 737 => [8, 9, 11, 12, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 29, 30, 31, 36] The maximum number of prime valued bases for base up to 62 numeric strings of length 3 is 33. The value list of these is: 1l1(base 32) => [22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62] b9b(base 32) => [13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 30, 31, 34, 36, 39, 40, 42, 45, 47, 49, 50, 52, 53, 54, 57, 58, 59, 60, 61, 62] The maximum number of prime valued bases for base up to 36 numeric strings of length 4 is 19. The value list of these is: 1727 => [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36] 5347 => [8, 9, 10, 11, 12, 13, 16, 18, 19, 22, 24, 25, 26, 30, 31, 32, 33, 34, 36] The maximum number of prime valued bases for base up to 62 numeric strings of length 4 is 32. The value list of these is: 1727 => [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36, 37, 38, 39, 41, 45, 46, 48, 50, 51, 57, 58, 60, 61] 417b(base 32) => [12, 13, 15, 16, 17, 18, 19, 21, 23, 25, 28, 30, 32, 34, 35, 37, 38, 39, 41, 45, 48, 49, 50, 51, 52, 54, 56, 57, 58, 59, 61, 62] The maximum number of prime valued bases for base up to 36 numeric strings of length 5 is 18. The value list of this is: 30271 => [8, 10, 12, 13, 16, 17, 18, 20, 21, 23, 24, 25, 31, 32, 33, 34, 35, 36] The maximum number of prime valued bases for base up to 62 numeric strings of length 5 is 30. The value list of this is: 50161 => [7, 8, 9, 13, 17, 18, 19, 20, 25, 28, 29, 30, 31, 33, 35, 36, 38, 39, 41, 42, 43, 44, 47, 48, 52, 55, 56, 59, 60, 62]
Mathematica/Wolfram Language
<lang Mathematica>ClearAll[OtherBasePrimes, OtherBasePrimesPower] OtherBasePrimes[n_Integer] := Module[{digs, minbase, bases},
digs = IntegerDigits[n]; minbase = Max[digs] + 1; bases = Range[minbase, 36]; Pick[bases, PrimeQ[FromDigits[digs, #] & /@ bases], True] ]
OtherBasePrimesPower[p_] := Module[{min, max, out, maxlen},
min = 10^p; max = 10^(p + 1) - 1; out = {#, OtherBasePrimes[#]} & /@ Range[min, max]; maxlen = Max[Length /@ outAll, 2]; Select[out, Last /* Length /* EqualTo[maxlen]] ]
OtherBasePrimesPower[0] // Column OtherBasePrimesPower[1] // Column OtherBasePrimesPower[2] // Column OtherBasePrimesPower[3] // Column OtherBasePrimesPower[4] // Column OtherBasePrimesPower[5] // Column</lang>
- Output:
{2,{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36}} {21,{3,5,6,8,9,11,14,15,18,20,21,23,26,29,30,33,35,36}} {131,{4,5,7,8,9,10,12,14,15,18,19,20,23,25,27,29,30,34}} {551,{6,7,11,13,14,15,16,17,19,21,22,24,25,26,30,32,35,36}} {737,{8,9,11,12,13,15,16,17,19,22,23,24,25,26,29,30,31,36}} {1727,{8,9,11,12,13,15,16,17,19,20,22,23,24,26,27,29,31,33,36}} {5347,{8,9,10,11,12,13,16,18,19,22,24,25,26,30,31,32,33,34,36}} {30271,{8,10,12,13,16,17,18,20,21,23,24,25,31,32,33,34,35,36}} {441431,{5,8,9,11,12,14,16,17,19,21,22,23,26,28,30,31,32,33}}
Nim
Compiled via C++ with full optimizations and runtime checks deactivated, the program takes 1 second to process up to 5 character strings and 34 seconds to process up to 6 characters (i5-8250U CPU @ 1.60GHz). Curiously, compiled via C it is slower (1.1 s and 38 seconds).
<lang Nim>import math, sequtils, strutils
const
MaxDepth = 6 Max = 36^MaxDepth - 1 # Max value for MaxDepth digits in base 36. Digits = "0123456789abcdefghijklmnopqrstuvwxyz"
- Sieve of Erathostenes.
var composite: array[1..(Max div 2), bool] # Only odd numbers. for i in 1..composite.high:
let n = 2 * i + 1 let n2 = n * n if n2 > Max: break if not composite[i]: for k in countup(n2, Max, 2 * n): composite[k shr 1] = true
template isPrime(n: int): bool =
if n <= 1: false elif (n and 1) == 0: n == 2 else: not composite[n shr 1]
type Context = object
indices: seq[int] mostBases: int maxStrings: seq[tuple[indices, bases: seq[int]]]
func initContext(depth: int): Context =
result.indices.setLen(depth) result.mostBases = -1
proc process(ctx: var Context) =
let minBase = max(2, max(ctx.indices) + 1) if 37 - minBase < ctx.mostBases: return
var bases: seq[int] for b in minBase..36: var n = 0 for i in ctx.indices: n = n * b + i if n.isPrime: bases.add b
var count = bases.len if count > ctx.mostBases: ctx.mostBases = count ctx.maxStrings = @{ctx.indices: bases} elif count == ctx.mostBases: ctx.maxStrings.add (ctx.indices, bases)
proc nestedFor(ctx: var Context; length, level: int) =
if level == ctx.indices.len: ctx.process() else: ctx.indices[level] = if level == 0: 1 else: 0 while ctx.indices[level] < length: ctx.nestedFor(length, level + 1) inc ctx.indices[level]
for depth in 1..MaxDepth:
var ctx = initContext(depth) ctx.nestedFor(Digits.len, 0) echo depth, " character strings which are prime in most bases: ", ctx.maxStrings[0].bases.len for m in ctx.maxStrings: echo m.indices.mapIt(Digits[it]).join(), " → ", m[1].join(" ") echo()</lang>
- Output:
1 character strings which are prime in most bases: 34 2 → 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 2 character strings which are prime in most bases: 18 21 → 3 5 6 8 9 11 14 15 18 20 21 23 26 29 30 33 35 36 3 character strings which are prime in most bases: 18 131 → 4 5 7 8 9 10 12 14 15 18 19 20 23 25 27 29 30 34 551 → 6 7 11 13 14 15 16 17 19 21 22 24 25 26 30 32 35 36 737 → 8 9 11 12 13 15 16 17 19 22 23 24 25 26 29 30 31 36 4 character strings which are prime in most bases: 19 1727 → 8 9 11 12 13 15 16 17 19 20 22 23 24 26 27 29 31 33 36 5347 → 8 9 10 11 12 13 16 18 19 22 24 25 26 30 31 32 33 34 36 5 character strings which are prime in most bases: 18 30271 → 8 10 12 13 16 17 18 20 21 23 24 25 31 32 33 34 35 36 6 character strings which are prime in most bases: 18 441431 → 5 8 9 11 12 14 16 17 19 21 22 23 26 28 30 31 32 33
Pascal
First counting the bases that convert a MAXBASE string of n into a prime number.
Afterwards only checking the maxcount for the used bases.
<lang pascal>program MAXBaseStringIsPrimeInBase;
{$IFDEF FPC}
{$MODE DELPHI} {$OPTIMIZATION ON,ALL}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF} uses
sysutils;
const
CharOfBase= '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz'; MINBASE = 2; MAXBASE = 62;//36;//62; MAXDIGITCOUNT = 5;//6;
type
tdigits = packed record dgtDgts : array [0..13] of byte; dgtMaxIdx, dgtMaxDgtVal :byte; dgtNum : Uint64; end; tSol = array of Uint64;
var
BoolPrimes: array of boolean;
function BuildWheel(primeLimit:Int64):NativeUint; var
myPrimes : pBoolean; wheelprimes :array[0..31] of byte; wheelSize,wpno, pr,pw,i, k: NativeUint;
begin
myPrimes := @BoolPrimes[0]; pr := 1; myPrimes[1]:= true; WheelSize := 1; wpno := 0; repeat inc(pr); pw := pr; if pw > wheelsize then dec(pw,wheelsize); If myPrimes[pw] then begin k := WheelSize+1; for i := 1 to pr-1 do begin inc(k,WheelSize); if k<primeLimit then move(myPrimes[1],myPrimes[k-WheelSize],WheelSize) else begin move(myPrimes[1],myPrimes[k-WheelSize],PrimeLimit-WheelSize*i); break; end; end; dec(k); IF k > primeLimit then k := primeLimit; wheelPrimes[wpno] := pr; myPrimes[pr] := false; inc(wpno); WheelSize := k; i:= pr; i := i*i; while i <= k do begin myPrimes[i] := false; inc(i,pr); end; end; until WheelSize >= PrimeLimit; while wpno > 0 do begin dec(wpno); myPrimes[wheelPrimes[wpno]] := true; end; myPrimes[0] := false; myPrimes[1] := false; BuildWheel := pr+1;
end;
procedure Sieve(PrimeLimit:Uint64); var
myPrimes : pBoolean; sieveprime, fakt : NativeUint;
begin
setlength(BoolPrimes,PrimeLimit+1);
myPrimes := @BoolPrimes[0]; sieveprime := BuildWheel(PrimeLimit); repeat if myPrimes[sieveprime] then begin fakt := PrimeLimit DIV sieveprime; IF fakt < sieveprime then BREAK; repeat myPrimes[sieveprime*fakt] := false; repeat dec(fakt); until myPrimes[fakt]; until fakt < sieveprime; end; inc(sieveprime); until false; myPrimes[1] := false;
end;
procedure CnvtoBASE(var dgt:tDigits;n:Uint64;base:NativeUint); var
q,r: Uint64; i : Int32;
Begin
i := 0; with dgt do Begin fillchar(dgtDgts,SizeOf(dgtDgts),#0); dgtNum:= n; repeat r := n; q := n div base; r -= q*base; n := q; dgtDgts[i] := r; inc(i); until (q = 0); dec(i); dgtMaxIdx := i; r := 1; repeat q := dgtDgts[i]; if r < q then r := q; dec(i); until i <0 ; dgtMaxDgtVal := r; end;
end;
function CnvDgtAsInBase(const dgt:tDigits;base:NativeUint):Uint64; var
tmpDgt,i: NativeInt;
Begin
result := 0; with dgt do Begin i:= dgtMaxIdx; repeat tmpDgt := dgtDgts[i]; result *= base; dec(i); result +=tmpDgt; until (i< 0); end;
end;
procedure IncInBaseDigits(var dgt:tDigits;base:NativeInt); var
i,q,tmp :NativeInt;
Begin
with dgt do Begin tmp := dgtMaxIdx; i := 0; repeat q := dgtDgts[i]+1; q -= (-ORD(q >=base) AND base); dgtDgts[i] := q; inc(i); until q <> 0; dec(i); if tmp < i then begin tmp := i; dgtMaxIdx := i; end; i := tmp; repeat tmp := dgtDgts[i]; if q< tmp then q := tmp; dec(i); until i <0; inc(dgtNum); dgtMaxDgtVal := q; end;
end;
function CntPrimeInBases(var Digits :tdigits;max:Int32):Uint32; var
pr : Uint64; base: Uint32;
begin
result := 0; IncInBaseDigits(Digits,MAXBASE); base := Digits.dgtMaxDgtVal+1; //divisible by every base IF Digits.dgtDgts[0] = 0 then EXIT; IF base < MINBASE then base := MINBASE;
// if (MAXBASE - Base) <= (max-result) then BREAK;
max := (max+Base-MAXBASE); if (max>=0) then EXIT; for base := base TO MAXBASE do begin pr := CnvDgtAsInBase(Digits,base); inc(result,Ord(boolprimes[pr])); //no chance to reach max then exit if result<max then break; inc(max); end;
end;
function GetMaxBaseCnt(var dgt:tDigits;MinLmt,MaxLmt:Uint32):tSol; var
i : Uint32; baseCnt,max,Idx: Int32;
Begin
setlength(result,0); max :=-1; Idx:= 0; For i := MinLmt to MaxLmt do Begin baseCnt := CntPrimeInBases(dgt,max); if baseCnt = 0 then continue; if max<=baseCnt then begin if max = baseCnt then begin inc(Idx); if Idx > High(result) then setlength(result,Idx); result[idx-1] := i; end else begin Idx:= 1; setlength(result,1); result[0] := i; max := baseCnt; end; end; end;
end;
function Out_String(n:Uint64;var s: AnsiString):Uint32; //out-sourced for debugging purpose var
dgt:tDigits; sl : string[15]; base,i: Int32;
Begin
result := 0; CnvtoBASE(dgt,n,MaxBase); sl := ; with dgt do begin base:= dgtMaxDgtVal+1; IF base < MINBASE then base := MINBASE; i := dgtMaxIdx; while (i>=0)do Begin sl += CharOfBase[dgtDgts[i]+1]; dec(i); end; s := sl+' -> ['; end; For base := base to MAXBASE do if boolprimes[CnvDgtAsInBase(dgt,base)] then begin inc(result); str(base,sl); s := s+sl+','; end; s[length(s)] := ']';
end;
procedure Out_Sol(sol:tSol); var
s : AnsiString; i,cnt : Int32;
begin
if length(Sol) = 0 then EXIT; for i := 0 to High(Sol) do begin cnt := Out_String(sol[i],s); if i = 0 then writeln(cnt); writeln(s); end; writeln; setlength(Sol,0);
end;
var
dgt:tDigits; T0 : Int64; i : NativeInt; lmt,minLmt : UInt64;
begin
T0 := GetTickCount64; lmt := 0; //maxvalue in Maxbase for i := 1 to MAXDIGITCOUNT do lmt :=lmt*MAXBASE+MAXBASE-1; writeln('max prime limit ',lmt); Sieve(lmt); writeln('Prime sieving ',(GetTickCount64-T0)/1000:6:3,' s');
T0 := GetTickCount64; CnvtoBASE(dgt,0,MAXBASE); i := 1; minLmt := 1; repeat write(i:2,' character strings which are prime in count bases = '); Out_Sol(GetMaxBaseCnt(dgt,minLmt,MAXBASE*minLmt-1)); minLmt *= MAXBASE; inc(i); until i>MAXDIGITCOUNT; writeln(' Converting ',(GetTickCount64-T0)/1000:6:3,' s'); {$IFDEF WINDOWS} readln; {$ENDIF}
end.</lang>
- Output:
TIO.RUN// extreme volatile timings for sieving primes max prime limit 916132831 Prime sieving 3.788 s 1 character strings which are prime in count bases = 60 2 -> [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62] 2 character strings which are prime in count bases = 31 65 -> [7,8,9,11,13,14,16,17,18,21,22,24,27,28,29,31,32,37,38,39,41,42,43,44,46,48,51,52,57,58,59] 3 character strings which are prime in count bases = 33 1L1 -> [22,23,25,26,27,28,29,30,31,32,33,34,36,38,39,40,41,42,43,44,45,46,48,51,52,53,54,57,58,59,60,61,62] B9B -> [13,14,15,16,17,19,20,21,23,24,26,27,28,30,31,34,36,39,40,42,45,47,49,50,52,53,54,57,58,59,60,61,62] 4 character strings which are prime in count bases = 32 1727 -> [8,9,11,12,13,15,16,17,19,20,22,23,24,26,27,29,31,33,36,37,38,39,41,45,46,48,50,51,57,58,60,61] 417B -> [12,13,15,16,17,18,19,21,23,25,28,30,32,34,35,37,38,39,41,45,48,49,50,51,52,54,56,57,58,59,61,62] 5 character strings which are prime in count bases = 30 50161 -> [7,8,9,13,17,18,19,20,25,28,29,30,31,33,35,36,38,39,41,42,43,44,47,48,52,55,56,59,60,62] Converting 12.738 s Real time: 16.768 s User time: 16.128 s Sys. time: 0.488 s CPU share: 99.09 % //at home AMD 2200G Linux fpc 3.2.2 real 0m8,609s user 0m8,378s sys 0m0,220s max prime limit 916132831 Prime sieving 1.734 s Converting 6.842 s //base = 36 maxcharacters = 6 max prime limit 2176782335 Prime sieving 4.986 s 1 character strings which are prime in count bases = 34 2 -> [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] 2 character strings which are prime in count bases = 18 21 -> [3,5,6,8,9,11,14,15,18,20,21,23,26,29,30,33,35,36] 3 character strings which are prime in count bases = 18 131 -> [4,5,7,8,9,10,12,14,15,18,19,20,23,25,27,29,30,34] 551 -> [6,7,11,13,14,15,16,17,19,21,22,24,25,26,30,32,35,36] 737 -> [8,9,11,12,13,15,16,17,19,22,23,24,25,26,29,30,31,36] 4 character strings which are prime in count bases = 19 1727 -> [8,9,11,12,13,15,16,17,19,20,22,23,24,26,27,29,31,33,36] 5347 -> [8,9,10,11,12,13,16,18,19,22,24,25,26,30,31,32,33,34,36] 5 character strings which are prime in count bases = 18 30271 -> [8,10,12,13,16,17,18,20,21,23,24,25,31,32,33,34,35,36] 6 character strings which are prime in count bases = 18 441431 -> [5,8,9,11,12,14,16,17,19,21,22,23,26,28,30,31,32,33] Converting 15.507 s// 24.3s before real 0m20,566s
Perl
<lang perl>use strict; use warnings; use feature 'say'; use List::AllUtils <firstidx max>; use ntheory qw/fromdigits todigitstring primes/;
my(%prime_base, %max_bases, $l);
my $chars = 5; my $upto = fromdigits( '1' . 'Z' x $chars, 36); my @primes = @{primes( $upto )};
for my $base (2..36) {
my $n = todigitstring($base-1, $base) x $chars; my $threshold = fromdigits($n, $base); my $i = firstidx { $_ > $threshold } @primes; map { push @{$prime_base{ todigitstring($primes[$_],$base) }}, $base } 0..$i-1;
}
$l = length and $max_bases{$l} = max( $#{$prime_base{$_}}, $max_bases{$l} // 0 ) for keys %prime_base;
for my $m (1 .. $chars) {
say "$m character strings that are prime in maximum bases: ", 1+$max_bases{$m}; for (sort grep { length($_) == $m } keys %prime_base) { my @bases = @{($prime_base{$_})[0]}; say "$_: " . join ' ', @bases if $max_bases{$m} eq $#bases; } say
}</lang>
- Output:
1 character strings that are prime in maximum bases: 34 2: 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 2 character strings that are prime in maximum bases: 18 21: 3 5 6 8 9 11 14 15 18 20 21 23 26 29 30 33 35 36 3 character strings that are prime in maximum bases: 18 131: 4 5 7 8 9 10 12 14 15 18 19 20 23 25 27 29 30 34 551: 6 7 11 13 14 15 16 17 19 21 22 24 25 26 30 32 35 36 737: 8 9 11 12 13 15 16 17 19 22 23 24 25 26 29 30 31 36 4 character strings that are prime in maximum bases: 19 1727: 8 9 11 12 13 15 16 17 19 20 22 23 24 26 27 29 31 33 36 5347: 8 9 10 11 12 13 16 18 19 22 24 25 26 30 31 32 33 34 36 5 character strings that are prime in maximum bases: 18 30271: 8 10 12 13 16 17 18 20 21 23 24 25 31 32 33 34 35 36
Phix
Originally translated from Rust, but changed to a much fuller range of digits, as per talk page.
with javascript_semantics constant maxbase=36 -- or 62 function evalpoly(integer x, sequence p) integer result = 0 for y=1 to length(p) do result = result*x + p[y] end for return result end function function stringify(sequence digits) string res = repeat('0',length(digits)) for i=1 to length(digits) do integer di = digits[i] res[i] = di + iff(di<=9?'0':iff(di<36?'A'-10:'a'-36)) end for return res end function procedure max_prime_bases(integer ndig, maxbase) atom t0 = time(), t1 = time()+1 sequence maxprimebases = {}, digits = repeat(0,ndig) integer maxlen = 0, limit = power(10,ndig), maxdigit = maxbase if ndig>1 then digits[1] = 1 end if while true do for i=length(digits) to 1 by -1 do integer di = digits[i]+1 if di<maxdigit then -- (or 9, see below) digits[i] = di exit else di = 0 digits[i] = 0 end if end for integer minbase = max(digits)+1, maxposs = maxbase-minbase+1 if minbase=1 then exit end if -- (ie we just wrapped round to all 0s) sequence bases = {} for base=minbase to maxbase do if is_prime(evalpoly(base,digits)) then bases &= base else maxposs -= 1 if maxposs<maxlen then exit end if -- (a 5-fold speedup) end if end for integer l = length(bases) if l>maxlen then maxlen = l maxdigit = maxbase-maxlen -- (around 20-fold speedup) maxprimebases = {} end if if l=maxlen then maxprimebases &= {{stringify(digits), bases}} end if if platform()!=JS and time()>t1 then progress("%V\r",{digits}) t1 = time()+1 end if end while string e = elapsed(time()-t0) printf(1,"%d character numeric strings that are prime in %d bases (%s):\n",{ndig,maxlen,e}) for i=1 to length(maxprimebases) do printf(1," %s => %V\n", maxprimebases[i]) end for printf(1,"\n") end procedure for n=1 to iff(platform()=JS or maxbase>36?4:6) do max_prime_bases(n, maxbase) end for
- Output:
1 character numeric strings that are prime in 34 bases (0s): 2 => {3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36} 2 character numeric strings that are prime in 18 bases (0s): 21 => {3,5,6,8,9,11,14,15,18,20,21,23,26,29,30,33,35,36} 3 character numeric strings that are prime in 18 bases (0.0s): 131 => {4,5,7,8,9,10,12,14,15,18,19,20,23,25,27,29,30,34} 551 => {6,7,11,13,14,15,16,17,19,21,22,24,25,26,30,32,35,36} 737 => {8,9,11,12,13,15,16,17,19,22,23,24,25,26,29,30,31,36} 4 character numeric strings that are prime in 19 bases (0.6s): 1727 => {8,9,11,12,13,15,16,17,19,20,22,23,24,26,27,29,31,33,36} 5347 => {8,9,10,11,12,13,16,18,19,22,24,25,26,30,31,32,33,34,36} 5 character numeric strings that are prime in 18 bases (18.6s): 30271 => {8,10,12,13,16,17,18,20,21,23,24,25,31,32,33,34,35,36} 6 character numeric strings that are prime in 18 bases (11 minutes and 17s): 441431 => {5,8,9,11,12,14,16,17,19,21,22,23,26,28,30,31,32,33}
As usual we skip the last couple of entries under pwa/p2js to avoid staring at a blank screen for ages
If we "cheat" and only check digits 0..9 we get the same results much faster:
4 character numeric strings that are prime in 19 bases (0.1s): 5 character numeric strings that are prime in 18 bases (1.0s): 6 character numeric strings that are prime in 18 bases (16.8s):
If we set maxbase to 62:
1 character numeric strings that are prime in 60 bases (0s): 2 => {3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62} 2 character numeric strings that are prime in 31 bases (0.0s): 65 => {7,8,9,11,13,14,16,17,18,21,22,24,27,28,29,31,32,37,38,39,41,42,43,44,46,48,51,52,57,58,59} 3 character numeric strings that are prime in 33 bases (0.2s): 1L1 => {22,23,25,26,27,28,29,30,31,32,33,34,36,38,39,40,41,42,43,44,45,46,48,51,52,53,54,57,58,59,60,61,62} B9B => {13,14,15,16,17,19,20,21,23,24,26,27,28,30,31,34,36,39,40,42,45,47,49,50,52,53,54,57,58,59,60,61,62} 4 character numeric strings that are prime in 32 bases (9.6s): 1727 => {8,9,11,12,13,15,16,17,19,20,22,23,24,26,27,29,31,33,36,37,38,39,41,45,46,48,50,51,57,58,60,61} 417B => {12,13,15,16,17,18,19,21,23,25,28,30,32,34,35,37,38,39,41,45,48,49,50,51,52,54,56,57,58,59,61,62}
Raku
Up to 4 character strings finish fairly quickly. 5 character strings take a while.
All your base are belong to us. You have no chance to survive make your prime. <lang perl6>use Math::Primesieve; my $sieve = Math::Primesieve.new;
my %prime-base;
my $chars = 4; # for demonstration purposes. Change to 5 for the whole shmegegge.
my $threshold = ('1' ~ 'Z' x $chars).parse-base(36);
my @primes = $sieve.primes($threshold);
%prime-base.push: $_ for (2..36).map: -> $base {
$threshold = (($base - 1).base($base) x $chars).parse-base($base); @primes[^(@primes.first: * > $threshold, :k)].race.map: { .base($base) => $base }
}
%prime-base.=grep: +*.value.elems > 10;
for 1 .. $chars -> $m {
say "$m character strings that are prime in maximum bases: " ~ (my $e = ((%prime-base.grep( *.key.chars == $m )).max: +*.value.elems).value.elems); .say for %prime-base.grep( +*.value.elems == $e ).grep(*.key.chars == $m).sort: *.key; say ;
}</lang>
- Output:
1 character strings that are prime in maximum bases: 34 2 => [3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36] 2 character strings that are prime in maximum bases: 18 21 => [3 5 6 8 9 11 14 15 18 20 21 23 26 29 30 33 35 36] 3 character strings that are prime in maximum bases: 18 131 => [4 5 7 8 9 10 12 14 15 18 19 20 23 25 27 29 30 34] 551 => [6 7 11 13 14 15 16 17 19 21 22 24 25 26 30 32 35 36] 737 => [8 9 11 12 13 15 16 17 19 22 23 24 25 26 29 30 31 36] 4 character strings that are prime in maximum bases: 19 1727 => [8 9 11 12 13 15 16 17 19 20 22 23 24 26 27 29 31 33 36] 5347 => [8 9 10 11 12 13 16 18 19 22 24 25 26 30 31 32 33 34 36] 5 character strings that are prime in maximum bases: 18 30271 => [8 10 12 13 16 17 18 20 21 23 24 25 31 32 33 34 35 36]
You can't really assume that the maximum string will be all numeric digits. It is just an accident that they happen to work out that way with a upper limit of base 36. If we do the same filtering using a maximum of base 62, we end up with several that contain alphabetics.
<lang perl6>use Math::Primesieve; use Base::Any;
my $chars = 4; my $check-base = 62; my $threshold = $check-base ** $chars + 20;
my $sieve = Math::Primesieve.new; my @primes = $sieve.primes($threshold);
my %prime-base;
%prime-base.push: $_ for (2..$check-base).map: -> $base {
$threshold = (($base - 1).&to-base($base) x $chars).&from-base($base); @primes[^(@primes.first: * > $threshold, :k)].race.map: { .&to-base($base) => $base }
}
%prime-base.=grep: +*.value.elems > 10;
for 1 .. $chars -> $m {
say "$m character strings that are prime in maximum bases: " ~ (my $e = ((%prime-base.grep( *.key.chars == $m )).max: +*.value.elems).value.elems); .say for %prime-base.grep( +*.value.elems == $e ).grep(*.key.chars == $m).sort: *.key; say ;
}</lang>
- Yields:
1 character strings that are prime in maximum bases: 60 2 => [3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62] 2 character strings that are prime in maximum bases: 31 65 => [7 8 9 11 13 14 16 17 18 21 22 24 27 28 29 31 32 37 38 39 41 42 43 44 46 48 51 52 57 58 59] 3 character strings that are prime in maximum bases: 33 1L1 => [22 23 25 26 27 28 29 30 31 32 33 34 36 38 39 40 41 42 43 44 45 46 48 51 52 53 54 57 58 59 60 61 62] B9B => [13 14 15 16 17 19 20 21 23 24 26 27 28 30 31 34 36 39 40 42 45 47 49 50 52 53 54 57 58 59 60 61 62] 4 character strings that are prime in maximum bases: 32 1727 => [8 9 11 12 13 15 16 17 19 20 22 23 24 26 27 29 31 33 36 37 38 39 41 45 46 48 50 51 57 58 60 61] 417B => [12 13 15 16 17 18 19 21 23 25 28 30 32 34 35 37 38 39 41 45 48 49 50 51 52 54 56 57 58 59 61 62]
REXX
<lang rexx>/*REXX pgm finds primes whose values in other bases (2──►36) have the most diff. bases. */ parse arg widths . /*obtain optional argument from the CL.*/ if widths== | widths=="," then widths= 5 /*Not specified? Then use the default.*/ call genP /*build array of semaphores for primes.*/ names= 'one two three four five six seven eight' /*names for some low decimal numbers. */ $.=
do j=1 for # /*only use primes that are within range*/ do b=36 by -1 for 35; n= base(@.j, b) /*use different bases for each prime. */ L= length(n); if L>widths then iterate /*obtain length; Length too big? Skip.*/ if L==1 then $.L.n= b $.L.n /*Length = unity? Prepend the base.*/ else $.L.n= $.L.n b /* " ¬= " Append " " */ end /*b*/ end /*j*/ /*display info for each of the widths. */ do w=1 for widths; cnt= 0 /*show for each width: cnt,number,bases*/ bot= left(1, w, 0); top= left(9, w, 9) /*calculate range for DO. */ do n=bot to top; y= words($.w.n) /*find the sets of numbers for a width.*/ if y>cnt then do; mxn=n; cnt= max(cnt, y); end /*found a max? Remember it*/ end /*n*/ say say; say center(' 'word(names, w)"─character numbers that are" , 'prime in the most bases: ('cnt "bases) ", 101, '─') do n=bot to top; y= words($.w.n) /*search again for maximums.*/ if y==cnt then say n '──►' strip($.w.n) /*display ───a─── maximum. */ end /*n*/ end /*w*/
exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ base: procedure; parse arg x,r,,z; @= '0123456789abcdefghijklmnopqrsruvwxyz'
do j=1; _= r**j; if _>x then leave end /*j*/ do k=j-1 to 1 by -1; _= r**k; z= z || substr(@, (x % _) + 1, 1); x= x // _ end /*k*/; return z || substr(@, x+1, 1)
/*──────────────────────────────────────────────────────────────────────────────────────*/ genP: @.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */
#= 5; sq.#= @.# ** 2 /*number primes so far; prime squared.*/ do j=@.#+2 by 2 to 2 * 36 * 10**widths /*find odd primes from here on. */ parse var j -1 _; if _==5 then iterate /*J is ÷ by 5? (right dig).*/ if j//3==0 then iterate; if j//7==0 then iterate /*" " " " 3?; ÷ by 7? */ do k=5 while sq.k<=j /* [↓] divide by the known odd primes.*/ if j//@.k==0 then iterate j /*Is J ÷ X? Then not prime. ___ */ end /*k*/ /* [↑] only process numbers ≤ √ J */ #= # + 1; @.#= j; sq.#= j*j /*bump # Ps; assign next P; P squared.*/ end /*j*/; return</lang>
- output when using the default input:
──────────────── one─character numbers that are prime in the most bases: (34 bases) ───────────────── 2 ──► 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ──────────────── two─character numbers that are prime in the most bases: (18 bases) ───────────────── 21 ──► 3 5 6 8 9 11 14 15 18 20 21 23 26 29 30 33 35 36 ─────────────── three─character numbers that are prime in the most bases: (18 bases) ──────────────── 131 ──► 4 5 7 8 9 10 12 14 15 18 19 20 23 25 27 29 30 34 551 ──► 6 7 11 13 14 15 16 17 19 21 22 24 25 26 30 32 35 36 737 ──► 8 9 11 12 13 15 16 17 19 22 23 24 25 26 29 30 31 36 ──────────────── four─character numbers that are prime in the most bases: (19 bases) ──────────────── 1727 ──► 8 9 11 12 13 15 16 17 19 20 22 23 24 26 27 29 31 33 36 5347 ──► 8 9 10 11 12 13 16 18 19 22 24 25 26 30 31 32 33 34 36 ──────────────── five─character numbers that are prime in the most bases: (18 bases) ──────────────── 30271 ──► 8 10 12 13 16 17 18 20 21 23 24 25 31 32 33 34 35 36
Rust
<lang rust>// [dependencies] // primal = "0.3"
fn digits(mut n: u32, dig: &mut [u32]) {
for i in 0..dig.len() { dig[i] = n % 10; n /= 10; }
}
fn evalpoly(x: u64, p: &[u32]) -> u64 {
let mut result = 0; for y in p.iter().rev() { result *= x; result += *y as u64; } result
}
fn max_prime_bases(ndig: u32, maxbase: u32) {
let mut maxlen = 0; let mut maxprimebases = Vec::new(); let limit = 10u32.pow(ndig); let mut dig = vec![0; ndig as usize]; for n in limit / 10..limit { digits(n, &mut dig); let bases: Vec<u32> = (2..=maxbase) .filter(|&x| dig.iter().all(|&y| y < x) && primal::is_prime(evalpoly(x as u64, &dig))) .collect(); if bases.len() > maxlen { maxlen = bases.len(); maxprimebases.clear(); } if bases.len() == maxlen { maxprimebases.push((n, bases)); } } println!( "{} character numeric strings that are prime in maximum bases: {}", ndig, maxlen ); for (n, bases) in maxprimebases { println!("{} => {:?}", n, bases); } println!();
}
fn main() {
for n in 1..=6 { max_prime_bases(n, 36); }
}</lang>
- Output:
1 character numeric strings that are prime in maximum bases: 34 2 => [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] 2 character numeric strings that are prime in maximum bases: 18 21 => [3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36] 3 character numeric strings that are prime in maximum bases: 18 131 => [4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 19, 20, 23, 25, 27, 29, 30, 34] 551 => [6, 7, 11, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 30, 32, 35, 36] 737 => [8, 9, 11, 12, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 29, 30, 31, 36] 4 character numeric strings that are prime in maximum bases: 19 1727 => [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36] 5347 => [8, 9, 10, 11, 12, 13, 16, 18, 19, 22, 24, 25, 26, 30, 31, 32, 33, 34, 36] 5 character numeric strings that are prime in maximum bases: 18 30271 => [8, 10, 12, 13, 16, 17, 18, 20, 21, 23, 24, 25, 31, 32, 33, 34, 35, 36] 6 character numeric strings that are prime in maximum bases: 18 441431 => [5, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 23, 26, 28, 30, 31, 32, 33]
Up to base 62
<lang rust>// [dependencies] // primal = "0.3"
fn to_string(digits: &[usize]) -> String {
const DIGITS: [char; 62] = [ '0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', 'x', 'y', 'z', 'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', 'X', 'Y', 'Z', ]; let mut str = String::new(); for d in digits { str.push(DIGITS[*d]); } str
}
fn increment(digits: &mut [usize], base: usize) -> bool {
for d in digits.iter_mut().rev() { if *d + 1 != base { *d += 1; return true; } *d = 0; } false
}
fn multi_base_primes(max_base: usize, max_length: usize) {
let sieve = primal::Sieve::new(max_base.pow(max_length as u32)); for length in 1..=max_length { let mut most_bases = 0; let mut max_strings = Vec::new(); let mut digits = vec![0; length]; digits[0] = 1; let mut bases = Vec::new(); loop { let mut min_base = 2; if let Some(max) = digits.iter().max() { min_base = std::cmp::max(min_base, max + 1); } if most_bases <= max_base - min_base + 1 { bases.clear(); for b in min_base..=max_base { if max_base - b + 1 + bases.len() < most_bases { break; } let mut n = 0; for d in &digits { n = n * b + d; } if sieve.is_prime(n) { bases.push(b); } } if bases.len() > most_bases { most_bases = bases.len(); max_strings.clear(); } if bases.len() == most_bases { max_strings.push((digits.clone(), bases.clone())); } } if !increment(&mut digits, max_base) { break; } } println!( "{}-character strings which are prime in most bases: {}", length, most_bases ); for (digits, bases) in max_strings { println!("{} -> {:?}", to_string(&digits), bases); } println!(); }
}
fn main() {
let args: Vec<String> = std::env::args().collect(); let mut max_base = 36; let mut max_length = 4; let mut arg = 0; while arg + 1 < args.len() { if args[arg] == "-max_base" { arg += 1; match args[arg].parse::<usize>() { Ok(n) => max_base = n, Err(error) => { eprintln!("{}", error); return; } } } else if args[arg] == "-max_length" { arg += 1; match args[arg].parse::<usize>() { Ok(n) => max_length = n, Err(error) => { eprintln!("{}", error); return; } } } arg += 1; } if max_base > 62 { eprintln!("Maximum base cannot be greater than 62."); } else if max_base < 2 { eprintln!("Maximum base cannot be less than 2."); } else { use std::time::Instant; let now = Instant::now(); multi_base_primes(max_base, max_length); let time = now.elapsed(); println!("elapsed time: {} milliseconds", time.as_millis()); }
}</lang>
- Output:
CPU: Intel Core i5-4570 3.2GHz. Maximum base 36, maximum length 6:
1-character strings which are prime in most bases: 34 2 -> [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] 2-character strings which are prime in most bases: 18 21 -> [3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36] 3-character strings which are prime in most bases: 18 131 -> [4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 19, 20, 23, 25, 27, 29, 30, 34] 551 -> [6, 7, 11, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 30, 32, 35, 36] 737 -> [8, 9, 11, 12, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 29, 30, 31, 36] 4-character strings which are prime in most bases: 19 1727 -> [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36] 5347 -> [8, 9, 10, 11, 12, 13, 16, 18, 19, 22, 24, 25, 26, 30, 31, 32, 33, 34, 36] 5-character strings which are prime in most bases: 18 30271 -> [8, 10, 12, 13, 16, 17, 18, 20, 21, 23, 24, 25, 31, 32, 33, 34, 35, 36] 6-character strings which are prime in most bases: 18 441431 -> [5, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 23, 26, 28, 30, 31, 32, 33] elapsed time: 15139 milliseconds
Maximum base 62, maximum length 5:
1-character strings which are prime in most bases: 60 2 -> [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] 2-character strings which are prime in most bases: 31 65 -> [7, 8, 9, 11, 13, 14, 16, 17, 18, 21, 22, 24, 27, 28, 29, 31, 32, 37, 38, 39, 41, 42, 43, 44, 46, 48, 51, 52, 57, 58, 59] 3-character strings which are prime in most bases: 33 1l1 -> [22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62] b9b -> [13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 30, 31, 34, 36, 39, 40, 42, 45, 47, 49, 50, 52, 53, 54, 57, 58, 59, 60, 61, 62] 4-character strings which are prime in most bases: 32 1727 -> [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36, 37, 38, 39, 41, 45, 46, 48, 50, 51, 57, 58, 60, 61] 417b -> [12, 13, 15, 16, 17, 18, 19, 21, 23, 25, 28, 30, 32, 34, 35, 37, 38, 39, 41, 45, 48, 49, 50, 51, 52, 54, 56, 57, 58, 59, 61, 62] 5-character strings which are prime in most bases: 30 50161 -> [7, 8, 9, 13, 17, 18, 19, 20, 25, 28, 29, 30, 31, 33, 35, 36, 38, 39, 41, 42, 43, 44, 47, 48, 52, 55, 56, 59, 60, 62] elapsed time: 9569 milliseconds
Sidef
<lang ruby>func max_prime_bases(ndig, maxbase=36) {
var maxprimebases = [[]] var nwithbases = [0] var maxprime = (10**ndig - 1)
for p in (idiv(maxprime + 1, 10) .. maxprime) { var dig = p.digits var bases = (2..maxbase -> grep {|b| dig.all { _ < b } && dig.digits2num(b).is_prime }) if (bases.len > maxprimebases.first.len) { maxprimebases = [bases] nwithbases = [p] } elsif (bases.len == maxprimebases.first.len) { maxprimebases << bases nwithbases << p } }
var (alen, vlen) = (maxprimebases.first.len, maxprimebases.len)
say("\nThe maximum number of prime valued bases for base 10 numeric strings of length ", ndig, " is #{alen}. The base 10 value list of ", vlen > 1 ? "these" : "this", " is:") maxprimebases.each_kv {|k,v| say(nwithbases[k], " => ", v) }
}
for n in (1..5) {
max_prime_bases(n)
}</lang>
- Output:
The maximum number of prime valued bases for base 10 numeric strings of length 1 is 34. The base 10 value list of this is: 2 => [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] The maximum number of prime valued bases for base 10 numeric strings of length 2 is 18. The base 10 value list of this is: 21 => [3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36] The maximum number of prime valued bases for base 10 numeric strings of length 3 is 18. The base 10 value list of these is: 131 => [4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 19, 20, 23, 25, 27, 29, 30, 34] 551 => [6, 7, 11, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 30, 32, 35, 36] 737 => [8, 9, 11, 12, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 29, 30, 31, 36] The maximum number of prime valued bases for base 10 numeric strings of length 4 is 19. The base 10 value list of these is: 1727 => [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36] 5347 => [8, 9, 10, 11, 12, 13, 16, 18, 19, 22, 24, 25, 26, 30, 31, 32, 33, 34, 36] The maximum number of prime valued bases for base 10 numeric strings of length 5 is 18. The base 10 value list of this is: 30271 => [8, 10, 12, 13, 16, 17, 18, 20, 21, 23, 24, 25, 31, 32, 33, 34, 35, 36]
Wren
This takes about 1.6 seconds to process up to 4 character strings and 58 seconds for the extra credit which is not too bad for the Wren interpreter. <lang ecmascript>import "/math" for Int, Nums
var maxDepth = 5 var maxBase = 36 var c = Int.primeSieve(maxBase.pow(maxDepth), false) var digits = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ" var maxStrings = [] var mostBases = -1
var process = Fn.new { |indices|
var minBase = 2.max(Nums.max(indices) + 1) if (maxBase - minBase + 1 < mostBases) return // can't affect results so return var bases = [] for (b in minBase..maxBase) { var n = 0 for (i in indices) n = n * b + i if (!c[n]) bases.add(b) } var count = bases.count if (count > mostBases) { mostBases = count maxStrings = indices.toList, bases } else if (count == mostBases) { maxStrings.add([indices.toList, bases]) }
}
var printResults = Fn.new {
System.print("%(maxStrings[0][1].count)") for (m in maxStrings) { var s = m[0].reduce("") { |acc, i| acc + digits[i] } System.print("%(s) -> %(m[1])") }
}
var nestedFor // recursive nestedFor = Fn.new { |indices, length, level|
if (level == indices.count) { process.call(indices) } else { indices[level] = (level == 0) ? 1 : 0 while (indices[level] < length) { nestedFor.call(indices, length, level + 1) indices[level] = indices[level] + 1 } }
}
for (depth in 1..maxDepth) {
System.write("%(depth) character strings which are prime in most bases: ") maxStrings = [] mostBases = -1 var indices = List.filled(depth, 0) nestedFor.call(indices, maxBase, 0) printResults.call() System.print()
}</lang>
- Output:
1 character strings which are prime in most bases: 34 2 -> [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] 2 character strings which are prime in most bases: 18 21 -> [3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36] 3 character strings which are prime in most bases: 18 131 -> [4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 19, 20, 23, 25, 27, 29, 30, 34] 551 -> [6, 7, 11, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 30, 32, 35, 36] 737 -> [8, 9, 11, 12, 13, 15, 16, 17, 19, 22, 23, 24, 25, 26, 29, 30, 31, 36] 4 character strings which are prime in most bases: 19 1727 -> [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36] 5347 -> [8, 9, 10, 11, 12, 13, 16, 18, 19, 22, 24, 25, 26, 30, 31, 32, 33, 34, 36] 5 character strings which are prime in most bases: 18 30271 -> [8, 10, 12, 13, 16, 17, 18, 20, 21, 23, 24, 25, 31, 32, 33, 34, 35, 36]
If we change maxBase to 62 and maxDepth to 4 in the above script, then the following results are reached in 17 seconds:
1 character strings which are prime in most bases: 60 2 -> [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] 2 character strings which are prime in most bases: 31 65 -> [7, 8, 9, 11, 13, 14, 16, 17, 18, 21, 22, 24, 27, 28, 29, 31, 32, 37, 38, 39, 41, 42, 43, 44, 46, 48, 51, 52, 57, 58, 59] 3 character strings which are prime in most bases: 33 1l1 -> [22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62] b9b -> [13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 30, 31, 34, 36, 39, 40, 42, 45, 47, 49, 50, 52, 53, 54, 57, 58, 59, 60, 61, 62] 4 character strings which are prime in most bases: 32 1727 -> [8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 31, 33, 36, 37, 38, 39, 41, 45, 46, 48, 50, 51, 57, 58, 60, 61] 417b -> [12, 13, 15, 16, 17, 18, 19, 21, 23, 25, 28, 30, 32, 34, 35, 37, 38, 39, 41, 45, 48, 49, 50, 51, 52, 54, 56, 57, 58, 59, 61, 62]