Multi-base primes

Prime numbers are prime no matter what base they are represented in.

Multi-base primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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++

Translation of: Wren

This takes 1.1 seconds to process up to 5 character strings and 40 seconds to process up to 6 characters (3.2GHz Intel Core i5). <lang cpp>#include <algorithm>

1. include <cmath>
2. include <cstdint>
3. include <iostream>
4. include <vector>

std::vector<bool> prime_sieve(uint64_t limit) {

   std::vector<bool> sieve(limit, true);
   if (limit > 0)
       sieve[0] = false;
   if (limit > 1)
       sieve[1] = false;
   for (uint64_t i = 4; i < limit; i += 2)
       sieve[i] = false;
   for (uint64_t p = 3;; p += 2) {
       uint64_t q = p * p;
       if (q >= limit)
           break;
       if (sieve[p]) {
           uint64_t inc = 2 * p;
           for (; q < limit; q += inc)
               sieve[q] = false;
       }
   }
   return sieve;

}

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[] = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ";
   std::string str;
   for (auto i : v)
       str += digits[i];
   return str;

};

class multi_base_primes { public:

   explicit multi_base_primes(unsigned int depth);
   void run();

private:

   void process(const std::vector<unsigned int>& indices);
   void nested_for(std::vector<unsigned int>& indices, unsigned int level);
   static const unsigned int max_base = 36;
   unsigned int max_depth;
   std::vector<bool> sieve;
   unsigned int most_bases = 0;
   std::vector<std::pair<std::vector<unsigned int>, std::vector<unsigned int>>>
       max_strings;

};

multi_base_primes::multi_base_primes(unsigned int depth)

   : max_depth(depth),
     sieve(prime_sieve(static_cast<uint64_t>(std::pow(max_base, depth)))) {}

void multi_base_primes::run() {

   for (unsigned int depth = 1; depth <= max_depth; ++depth) {
       std::cout << depth
                 << " character strings which are prime in most bases: ";
       max_strings.clear();
       most_bases = 0;
       std::vector<unsigned int> indices(depth, 0);
       nested_for(indices, 0);
       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';
   }

}

void multi_base_primes::process(const std::vector<unsigned int>& indices) {

   auto max = std::max_element(indices.begin(), indices.end());
   unsigned int min_base = 2;
   if (max != indices.end())
       min_base = std::max(min_base, *max + 1);
   if (most_bases > max_base - min_base)
       return;
   std::vector<unsigned int> bases;
   for (unsigned int b = min_base; b <= max_base; ++b) {
       uint64_t n = 0;
       for (auto i : indices)
           n = n * b + i;
       if (sieve[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(indices, bases);

}

void multi_base_primes::nested_for(std::vector<unsigned int>& indices,

                                  unsigned int level) {
   if (level == indices.size()) {
       process(indices);
   } else {
       indices[level] = (level == 0) ? 1 : 0;
       while (indices[level] < max_base) {
           nested_for(indices, level + 1);
           ++indices[level];
       }
   }

}

int main() {

   multi_base_primes mbp(6);
   mbp.run();

}</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]

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

Works with: Factor version 0.99 2021-06-02

<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

Translation of: Wren

Library: Go-rcu

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 c = rcu.PrimeSieve(int(math.Pow(36, float64(maxDepth))), true) var digits = "0123456789abcdefghijklmnopqrstuvwxyz" 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 37 - minBase < mostBases {
       return  // can't affect results so return
   }
   var bases []int
   for b := minBase; b <= 36; 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, len(digits), 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]

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)

Pascal

First counting the bases that convert a decimal string of n into a prime number.
Afterwards only checking the maxcount for the used bases.
Most time consuming is sieving for the primes. <lang pascal>program DecStringIsPrimeInBase; //base 10 numeric string {$IFDEF FPC}

 {$MODE DELPHI}
 {$OPTIMIZATION ON,ALL}
 //{$R+,O+}

{$ELSE}

 {$APPTYPE CONSOLE}

{$ENDIF} uses

 sysutils;

const

 MINBASE = 2;
 MAXBASE = 36;//10;//10 is minimum
 MAXDIGITCOUNT = 6;//9;//
 MAXFAC  = 10*10*10*10*10*10;//10*10*10*10*10*10*10*10*10;//

type

 tdigits    = array [0..15] of byte;//must be 0..15
 tSol       = array of Uint32;

//sieve primes see http://rosettacode.org/wiki/Sieve_of_Eratosthenes#alternative_using_wheel var

 BoolPrimes: array  of boolean;

function BuildWheel(primeLimit:Int32):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:Int32); 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;

function getDecDigitsAndMaxDgt(n:Uint32;var dgt:tDigits):uint32; var

 pQ :pQWord;
 q,r,i: Uint32;

Begin

 pQ := @dgt[0];pQ[0] := 0;pQ[1] := 0;

//aka fillChar(dgt[0],SizeOf(dgt),#0);

 i := 0;
 result := 0;
 repeat
   q := n DIV 10;
   r := (n-q*10);
   n := q;
   dgt[i] := r;
   inc(i);
   if result < r then
     result := r;
 until n = 0;

end;

function CnvtoBase(const dgt:tDigits;base:Uint32):Uint32; var

 i: Int32;

Begin

 i := MAXDIGITCOUNT;
 while (dgt[i] = 0) AND (i>0) do
   dec(i);

 result := 0;
 repeat
   result := base*result+dgt[i];
   dec(i);
 until (i< 0);

end;

function CntPrimeInBases(n:Uint32):Uint32; var

 Digits :tdigits;
 base: Uint32;

begin

 //base 10
 result := Ord(boolprimes[n]);
 //minimalBase <= max. digit +1
 //this takes 0.012s out of 0.155s
 base := getDecDigitsAndMaxDgt(n,Digits)+1;
 if base < MinBase then
   base := MinBase;

 for base := base TO 9 do
   inc(result,Ord(boolprimes[CnvtoBase(Digits,base)]));
 for base := 11 TO MAXBASE do
   inc(result,Ord(boolprimes[CnvtoBase(Digits,base)]));

end;

function GetMaxBaseCnt(MinLmt,MaxLmt:Uint32):tSol; var

 i,baseCnt,max,Idx: Int32;

Begin

 setlength(result,0);
 max :=-1;
 Idx:= 0;
 For i := MinLmt to MaxLmt do
 Begin
   baseCnt := CntPrimeInBases(i);
   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:Uint32;var s: AnsiString):Uint32; //out-sourced for debugging purpose var

 dgt:tDigits;
 sl : string[8];
 base,minimalbase: Uint32;

Begin

 result := 0;
 str(n:7,sl);
 s := sl+' -> [';
 minimalbase:= getDecDigitsAndMaxDgt(n,dgt)+1;
 For base := minimalbase to MAXBASE do
   if boolprimes[CnvtoBase(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

 T0 : Int64;
 i,lmt,minLmt : Uint32;

begin

 T0 := GetTickCount64;
 lmt := 0;
 //maxvalue of "99...99" in Maxbase
 for i := 1 to MAXDIGITCOUNT do
   lmt := (lmt*MAXBASE+9);
 writeln('max prime limit ',lmt);
 Sieve(lmt);
 writeln('Prime sieving ',(GetTickCount64-T0)/1000:6:3,' s');
 T0 := GetTickCount64;
 i := 1;
 minLmt := 1;
 repeat
   write(i:2,' character strings which are prime in count bases = ');
   Out_Sol(GetMaxBaseCnt(minLmt,10*minLmt-1));
   minLmt *= 10;
   inc(i);
 until minLmt >= MAXFAC;
 writeln('  Converting ',(GetTickCount64-T0)/1000:6:3,' s');
 {$IFDEF WINDOWS} readln; {$ENDIF}

end.</lang>

Output:

TIO.RUN// extreme volatile timings for sieving primes:today up to 15.6s
max prime limit 559744029
Prime sieving  2.168 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  0.333 s
//at home
//Prime sieving  1.071 s  Converting  0.171 s

Phix

Originally translated from Rust, but changed to a much fuller range of digits, as per talk page.

with javascript_semantics 
 
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':'a'-10)
    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?4:6) do
    max_prime_bases(n, 36)
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):

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]

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;  Lenth 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($.w.n)               /*search again for maxes. */
         if y==cnt  then say n  '──►'  strip($.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;   s.#= @.# **2   /*number of primes so far;     prime². */
       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 divisible by 5?  (right dig)*/
                            if j// 3==0  then iterate  /*"     "      " 3?             */
                            if j// 7==0  then iterate  /*"     "      " 7?             */
              do k=5  while s.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;    s.#= j*j           /*bump # of Ps; assign next P; P square*/
       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

Translation of: Julia

<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]

Wren

Library: Wren-math

Library: Wren-fmt

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 import "/fmt" for Conv

var maxDepth = 5 var c = Int.primeSieve(36.pow(maxDepth), false) var digits = Conv.digits // all digits up to base 36 var maxStrings = [] var mostBases = -1

var process = Fn.new { |indices|

   var minBase = 2.max(Nums.max(indices) + 1)
   if (37 - minBase < mostBases) return  // can't affect results so return
   var bases = []
   for (b in minBase..36) {
       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, digits.count, 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]