Repunit primes

Repunit is a portmanteau of the words "repetition" and "unit", with unit being "unit value"... or in laymans terms, 1. So 1, 11, 111, 1111 & 11111 are all repunits.

Task
Repunit primes
You are encouraged to solve this task according to the task description, using any language you may know.

Every standard integer base has repunits since every base has the digit 1. This task involves finding the repunits in different bases that are prime.

In base two, the repunits 11, 111, 11111, 1111111, etc. are prime. (These correspond to the Mersenne primes.)

In base three: 111, 1111111, 1111111111111, etc.

Repunit primes, by definition, are also circular primes.

Any repunit in any base having a composite number of digits is necessarily composite. Only repunits (in any base) having a prime number of digits might be prime.


Rather than expanding the repunit out as a giant list of 1s or converting to base 10, it is common to just list the number of 1s in the repunit; effectively the digit count. The base two repunit primes listed above would be represented as: 2, 3, 5, 7, etc.

Many of these sequences exist on OEIS, though they aren't specifically listed as "repunit prime digits" sequences.

Some bases have very few repunit primes. Bases 4, 8, and likely 16 have only one. Base 9 has none at all. Bases above 16 may have repunit primes as well... but this task is getting large enough already.


Task
  • For bases 2 through 16, Find and show, here on this page, the repunit primes as digit counts, up to a limit of 1000.


Stretch
  • Increase the limit to 2700 (or as high as you have patience for.)


See also


C++Edit

Library: GMP
Library: Primesieve
#include <future>
#include <iomanip>
#include <iostream>
#include <vector>

#include <gmpxx.h>
#include <primesieve.hpp>

std::vector<uint64_t> repunit_primes(uint32_t base,
                                     const std::vector<uint64_t>& primes) {
    std::vector<uint64_t> result;
    for (uint64_t prime : primes) {
        mpz_class repunit(std::string(prime, '1'), base);
        if (mpz_probab_prime_p(repunit.get_mpz_t(), 25) != 0)
            result.push_back(prime);
    }
    return result;
}

int main() {
    std::vector<uint64_t> primes;
    const uint64_t limit = 2700;
    primesieve::generate_primes(limit, &primes);
    std::vector<std::future<std::vector<uint64_t>>> futures;
    for (uint32_t base = 2; base <= 36; ++base) {
        futures.push_back(std::async(repunit_primes, base, primes));
    }
    std::cout << "Repunit prime digits (up to " << limit << ") in:\n";
    for (uint32_t base = 2, i = 0; base <= 36; ++base, ++i) {
        std::cout << "Base " << std::setw(2) << base << ':';
        for (auto digits : futures[i].get())
            std::cout << ' ' << digits;
        std::cout << '\n';
    }
}
Output:

This takes about 4 minutes 12 seconds (3.2GHz Quad-Core Intel Core i5).

Repunit prime digits (up to 2700) in:
Base  2: 2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281
Base  3: 3 7 13 71 103 541 1091 1367 1627
Base  4: 2
Base  5: 3 7 11 13 47 127 149 181 619 929
Base  6: 2 3 7 29 71 127 271 509 1049
Base  7: 5 13 131 149 1699
Base  8: 3
Base  9:
Base 10: 2 19 23 317 1031
Base 11: 17 19 73 139 907 1907 2029
Base 12: 2 3 5 19 97 109 317 353 701
Base 13: 5 7 137 283 883 991 1021 1193
Base 14: 3 7 19 31 41 2687
Base 15: 3 43 73 487 2579
Base 16: 2
Base 17: 3 5 7 11 47 71 419
Base 18: 2
Base 19: 19 31 47 59 61 107 337 1061
Base 20: 3 11 17 1487
Base 21: 3 11 17 43 271
Base 22: 2 5 79 101 359 857
Base 23: 5
Base 24: 3 5 19 53 71 653 661
Base 25:
Base 26: 7 43 347
Base 27: 3
Base 28: 2 5 17 457 1423
Base 29: 5 151
Base 30: 2 5 11 163 569 1789
Base 31: 7 17 31
Base 32:
Base 33: 3 197
Base 34: 13 1493
Base 35: 313 1297
Base 36: 2

F#Edit

This task uses Extensible Prime Generator (F#)

// Repunit primes. Nigel Galloway: January 24th., 2022
let rUnitP(b:int)=let b=bigint b in primes32()|>Seq.takeWhile((>)1000)|>Seq.map(fun n->(n,((b**n)-1I)/(b-1I)))|>Seq.filter(fun(_,n)->Open.Numeric.Primes.MillerRabin.IsProbablePrime &n)|>Seq.map fst
[2..16]|>List.iter(fun n->printf $"Base %d{n}: "; rUnitP(n)|>Seq.iter(printf "%d "); printfn "")
Output:
Base 2: 2 3 5 7 13 17 19 31 61 89 107 127 521 607
Base 3: 3 7 13 71 103 541
Base 4: 2
Base 5: 3 7 11 13 47 127 149 181 619 929
Base 6: 2 3 7 29 71 127 271 509
Base 7: 5 13 131 149
Base 8: 3
Base 9:
Base 10: 2 19 23 317
Base 11: 17 19 73 139 907
Base 12: 2 3 5 19 97 109 317 353 701
Base 13: 5 7 137 283 883 991
Base 14: 3 7 19 31 41
Base 15: 3 43 73 487
Base 16: 2

GoEdit

Translation of: Wren
Library: Go-rcu

Took 11 minutes 25 seconds which is much the same as Wren's GMP wrapper.

package main

import (
    "fmt"
    big "github.com/ncw/gmp"
    "rcu"
    "strings"
)

func main() {
    limit := 2700
    primes := rcu.Primes(limit)
    s := new(big.Int)
    for b := 2; b <= 36; b++ {
        var rPrimes []int
        for _, p := range primes {
            s.SetString(strings.Repeat("1", p), b)
            if s.ProbablyPrime(15) {
                rPrimes = append(rPrimes, p)
            }
        }
        fmt.Printf("Base %2d: %v\n", b, rPrimes)
    }
}
Output:
Base  2: [2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281]
Base  3: [3 7 13 71 103 541 1091 1367 1627]
Base  4: [2]
Base  5: [3 7 11 13 47 127 149 181 619 929]
Base  6: [2 3 7 29 71 127 271 509 1049]
Base  7: [5 13 131 149 1699]
Base  8: [3]
Base  9: []
Base 10: [2 19 23 317 1031]
Base 11: [17 19 73 139 907 1907 2029]
Base 12: [2 3 5 19 97 109 317 353 701]
Base 13: [5 7 137 283 883 991 1021 1193]
Base 14: [3 7 19 31 41 2687]
Base 15: [3 43 73 487 2579]
Base 16: [2]
Base 17: [3 5 7 11 47 71 419]
Base 18: [2]
Base 19: [19 31 47 59 61 107 337 1061]
Base 20: [3 11 17 1487]
Base 21: [3 11 17 43 271]
Base 22: [2 5 79 101 359 857]
Base 23: [5]
Base 24: [3 5 19 53 71 653 661]
Base 25: []
Base 26: [7 43 347]
Base 27: [3]
Base 28: [2 5 17 457 1423]
Base 29: [5 151]
Base 30: [2 5 11 163 569 1789]
Base 31: [7 17 31]
Base 32: []
Base 33: [3 197]
Base 34: [13 1493]
Base 35: [313 1297]
Base 36: [2]

JuliaEdit

using Primes

repunitprimeinbase(n, base) = isprime(evalpoly(BigInt(base), [1 for _ in 1:n]))

for b in 2:40
    println(rpad("Base $b:", 9), filter(n -> repunitprimeinbase(n, b), 1:2700))
end
Output:
Base 2:  [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281]
Base 3:  [3, 7, 13, 71, 103, 541, 1091, 1367, 1627]
Base 4:  [2]
Base 5:  [3, 7, 11, 13, 47, 127, 149, 181, 619, 929]
Base 6:  [2, 3, 7, 29, 71, 127, 271, 509, 1049]
Base 7:  [5, 13, 131, 149, 1699]
Base 8:  [3]
Base 9:  Int64[]
Base 10: [2, 19, 23, 317, 1031]
Base 11: [17, 19, 73, 139, 907, 1907, 2029]
Base 12: [2, 3, 5, 19, 97, 109, 317, 353, 701]
Base 13: [5, 7, 137, 283, 883, 991, 1021, 1193]
Base 14: [3, 7, 19, 31, 41, 2687]
Base 15: [3, 43, 73, 487, 2579]
Base 16: [2]
Base 17: [3, 5, 7, 11, 47, 71, 419]
Base 18: [2]
Base 19: [19, 31, 47, 59, 61, 107, 337, 1061]
Base 20: [3, 11, 17, 1487]
Base 21: [3, 11, 17, 43, 271]
Base 22: [2, 5, 79, 101, 359, 857]
Base 23: [5]
Base 24: [3, 5, 19, 53, 71, 653, 661]
Base 25: Int64[]
Base 26: [7, 43, 347]
Base 27: [3]
Base 28: [2, 5, 17, 457, 1423]
Base 29: [5, 151]
Base 30: [2, 5, 11, 163, 569, 1789]
Base 31: [7, 17, 31]
Base 32: Int64[]
Base 33: [3, 197]
Base 34: [13, 1493]
Base 35: [313, 1297]
Base 36: [2]
Base 37: [13, 71, 181, 251, 463, 521]
Base 38: [3, 7, 401, 449]
Base 39: [349, 631]
Base 40: [2, 5, 7, 19, 23, 29, 541, 751, 1277]

Mathematica/Wolfram LanguageEdit

ClearAll[RepUnitPrimeQ]
RepUnitPrimeQ[b_][n_] := PrimeQ[FromDigits[ConstantArray[1, n], b]]
ClearAll[RepUnitPrimeQ]
RepUnitPrimeQ[b_][n_] := PrimeQ[FromDigits[ConstantArray[1, n], b]]
Do[
 Print["Base ", b, ": ", Select[Range[2700], RepUnitPrimeQ[b]]]
 ,
 {b, 2, 16}
]

Searching bases 2–16 for repunits of size 2700:

Output:
Base 2: {2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281}
Base 3: {3,7,13,71,103,541,1091,1367,1627}
Base 4: {2}
Base 5: {3,7,11,13,47,127,149,181,619,929}
Base 6: {2,3,7,29,71,127,271,509,1049}
Base 7: {5,13,131,149,1699}
Base 8: {3}
Base 9: {}
Base 10: {2,19,23,317,1031}
Base 11: {17,19,73,139,907,1907,2029}
Base 12: {2,3,5,19,97,109,317,353,701}
Base 13: {5,7,137,283,883,991,1021,1193}
Base 14: {3,7,19,31,41,2687}
Base 15: {3,43,73,487,2579}
Base 16: {2}

PerlEdit

Library: ntheory
use strict;
use warnings;
use ntheory <is_prime fromdigits>;

my $limit = 1000;

print "Repunit prime digits (up to $limit) in:\n";

for my $base (2..16) {
    printf "Base %2d: %s\n", $base, join ' ', grep { is_prime $_ and is_prime fromdigits(('1'x$_), $base) and " $_" } 1..$limit
}
Output:
Repunit prime digits (up to 1000) in:
Base  2: 2 3 5 7 13 17 19 31 61 89 107 127 521 607
Base  3: 3 7 13 71 103 541
Base  4: 2
Base  5: 3 7 11 13 47 127 149 181 619 929
Base  6: 2 3 7 29 71 127 271 509
Base  7: 5 13 131 149
Base  8: 3
Base  9:
Base 10: 2 19 23 317
Base 11: 17 19 73 139 907
Base 12: 2 3 5 19 97 109 317 353 701
Base 13: 5 7 137 283 883 991
Base 14: 3 7 19 31 41
Base 15: 3 43 73 487
Base 16: 2

PhixEdit

with javascript_semantics
include mpfr.e
procedure repunit(mpz z, integer n, base=10)
    mpz_set_si(z,0)
    for i=1 to n do
        mpz_mul_si(z,z,base)
        mpz_add_si(z,z,1)
    end for
end procedure 
 
atom t0 = time()
constant {limit,blimit} = iff(platform()=JS?{400,16} -- 8.8s
                                           :{1000,16}) -- 50.3s
--                                         :{1000,36}) -- 4 min 20s
--                                         :{2700,16}) -- 28 min 35s
--                                         :{2700,36}) -- >patience
sequence primes = get_primes_le(limit)
mpz z = mpz_init() 
for base=2 to blimit do
    sequence rprimes = {}
    for i=1 to length(primes) do
        integer p = primes[i]
        repunit(z,p,base)
        if mpz_prime(z) then
            rprimes = append(rprimes,sprint(p))
        end if
    end for
    printf(1,"Base %2d: %s\n", {base, join(rprimes)})
end for
?elapsed(time()-t0)
Output:

Note the first half (base<=16) is from a {2700,16} run and the rest (base>16) from a {1000,36} run.

Base  2: 2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281
Base  3: 3 7 13 71 103 541 1091 1367 1627
Base  4: 2
Base  5: 3 7 11 13 47 127 149 181 619 929
Base  6: 2 3 7 29 71 127 271 509 1049
Base  7: 5 13 131 149 1699
Base  8: 3
Base  9:
Base 10: 2 19 23 317 1031
Base 11: 17 19 73 139 907 1907 2029
Base 12: 2 3 5 19 97 109 317 353 701
Base 13: 5 7 137 283 883 991 1021 1193
Base 14: 3 7 19 31 41 2687
Base 15: 3 43 73 487 2579
Base 16: 2
Base 17: 3 5 7 11 47 71 419
Base 18: 2
Base 19: 19 31 47 59 61 107 337
Base 20: 3 11 17
Base 21: 3 11 17 43 271
Base 22: 2 5 79 101 359 857
Base 23: 5
Base 24: 3 5 19 53 71 653 661
Base 25:
Base 26: 7 43 347
Base 27: 3
Base 28: 2 5 17 457
Base 29: 5 151
Base 30: 2 5 11 163 569
Base 31: 7 17 31
Base 32:
Base 33: 3 197
Base 34: 13
Base 35: 313
Base 36: 2

PythonEdit

from sympy import isprime
for b in range(2, 17):
    print(b, [n for n in range(2, 1001) if isprime(n) and isprime(int('1'*n, base=b))])
Output:
2 [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607]
3 [3, 7, 13, 71, 103, 541]
4 [2]
5 [3, 7, 11, 13, 47, 127, 149, 181, 619, 929]
6 [2, 3, 7, 29, 71, 127, 271, 509]
7 [5, 13, 131, 149]
8 [3]
9 []
10 [2, 19, 23, 317]
11 [17, 19, 73, 139, 907]
12 [2, 3, 5, 19, 97, 109, 317, 353, 701]
13 [5, 7, 137, 283, 883, 991]
14 [3, 7, 19, 31, 41]
15 [3, 43, 73, 487]
16 [2]

RakuEdit

my $limit = 2700;

say "Repunit prime digits (up to $limit) in:";

.put for (2..16).hyper(:1batch).map: -> $base {
    $base.fmt("Base %2d: ") ~ (1..$limit).grep(&is-prime).grep( (1 x *).parse-base($base).is-prime )
}
Output:
Repunit prime digits (up to 2700) in:
Base  2: 2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281
Base  3: 3 7 13 71 103 541 1091 1367 1627
Base  4: 2
Base  5: 3 7 11 13 47 127 149 181 619 929
Base  6: 2 3 7 29 71 127 271 509 1049
Base  7: 5 13 131 149 1699
Base  8: 3
Base  9: 
Base 10: 2 19 23 317 1031
Base 11: 17 19 73 139 907 1907 2029
Base 12: 2 3 5 19 97 109 317 353 701
Base 13: 5 7 137 283 883 991 1021 1193
Base 14: 3 7 19 31 41 2687
Base 15: 3 43 73 487 2579
Base 16: 2

And for my own amusement, also tested up to 2700.

Base 17: 3 5 7 11 47 71 419
Base 18: 2
Base 19: 19 31 47 59 61 107 337 1061
Base 20: 3 11 17 1487
Base 21: 3 11 17 43 271
Base 22: 2 5 79 101 359 857
Base 23: 5
Base 24: 3 5 19 53 71 653 661
Base 25: 
Base 26: 7 43 347
Base 27: 3
Base 28: 2 5 17 457 1423
Base 29: 5 151
Base 30: 2 5 11 163 569 1789
Base 31: 7 17 31
Base 32: 
Base 33: 3 197
Base 34: 13 1493
Base 35: 313 1297
Base 36: 2

SchemeEdit

Works with: Chez Scheme

This uses Miller–Rabin primality test (Scheme) , renamed, restructured, with a minor fix.
Took about 59 minutes to run.
Primality Test

; Test whether any integer is a probable prime.
(define prime<probably>?
  (lambda (n)
    ; Fast modular exponentiation.
    (define modexpt
      (lambda (b e m)
        (cond
          ((zero? e) 1)
          ((even? e) (modexpt (mod (* b b) m) (div e 2) m))
          ((odd? e) (mod (* b (modexpt b (- e 1) m)) m)))))
    ; Return multiple values s, d such that d is odd and 2^s * d = n.
    (define split
      (lambda (n)
        (let recur ((s 0) (d n))
          (if (odd? d)
            (values s d)
            (recur (+ s 1) (div d 2))))))
    ; Test whether the number a proves that n is composite.
    (define composite-witness?
      (lambda (n a)
        (let*-values (((s d) (split (- n 1)))
                      ((x) (modexpt a d n)))
          (and (not (= x 1))
               (not (= x (- n 1)))
               (let try ((r (- s 1)))
                 (set! x (modexpt x 2 n))
                 (or (zero? r)
                     (= x 1)
                     (and (not (= x (- n 1)))
                          (try (- r 1)))))))))
    ; Test whether n > 2 is a Miller-Rabin pseudoprime, k trials.
    (define pseudoprime?
      (lambda (n k)
        (or (zero? k)
            (let ((a (+ 2 (random (- n 2)))))
              (and (not (composite-witness? n a))
                   (pseudoprime? n (- k 1)))))))
    ; Compute and return Probable Primality using the Miller-Rabin algorithm.
    (and (> n 1)
         (or (= n 2)
             (and (odd? n)
                  (pseudoprime? n 50))))))

The Task

; Return list of the Repunit Primes in the given base up to the given limit.
(define repunit_primes
  (lambda (base limit)
    (let loop ((count 2)
               (value (1+ base)))
      (cond ((> count limit)
              '())
            ((and (prime<probably>? count) (prime<probably>? value))
              (cons count (loop (1+ count) (+ value (expt base count)))))
            (else
              (loop (1+ count) (+ value (expt base count))))))))

; Show all the Repunit Primes up to 2700 digits for bases 2 through 16.
(let ((max-base 16)
      (max-digits 2700))
  (printf "~%Repunit Primes up to ~d digits for bases 2 through ~d:~%" max-digits max-base)
  (do ((base 2 (1+ base)))
      ((> base max-base))
    (printf "Base ~2d: ~a~%" base (repunit_primes base max-digits))))
Output:
Repunit Primes up to 2700 digits for bases 2 through 16:
Base  2: (2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281)
Base  3: (3 7 13 71 103 541 1091 1367 1627)
Base  4: (2)
Base  5: (3 7 11 13 47 127 149 181 619 929)
Base  6: (2 3 7 29 71 127 271 509 1049)
Base  7: (5 13 131 149 1699)
Base  8: (3)
Base  9: ()
Base 10: (2 19 23 317 1031)
Base 11: (17 19 73 139 907 1907 2029)
Base 12: (2 3 5 19 97 109 317 353 701)
Base 13: (5 7 137 283 883 991 1021 1193)
Base 14: (3 7 19 31 41 2687)
Base 15: (3 43 73 487 2579)
Base 16: (2)

SidefEdit

var limit = 1000

say "Repunit prime digits (up to #{limit}) in:"

for n in (2..20) {
    printf("Base %2d: %s\n", n,
        {|k| is_prime((n**k - 1) / (n-1)) }.grep(1..limit))
}
Output:
Base  2: [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607]
Base  3: [3, 7, 13, 71, 103, 541]
Base  4: [2]
Base  5: [3, 7, 11, 13, 47, 127, 149, 181, 619, 929]
Base  6: [2, 3, 7, 29, 71, 127, 271, 509]
Base  7: [5, 13, 131, 149]
Base  8: [3]
Base  9: []
Base 10: [2, 19, 23, 317]
Base 11: [17, 19, 73, 139, 907]
Base 12: [2, 3, 5, 19, 97, 109, 317, 353, 701]
Base 13: [5, 7, 137, 283, 883, 991]
Base 14: [3, 7, 19, 31, 41]
Base 15: [3, 43, 73, 487]
Base 16: [2]
Base 17: [3, 5, 7, 11, 47, 71, 419]
Base 18: [2]
Base 19: [19, 31, 47, 59, 61, 107, 337]
Base 20: [3, 11, 17]

WrenEdit

Library: Wren-gmp
Library: Wren-math
Library: Wren-fmt
Library: Wren-str

An embedded program so we can use GMP as I know from experience that Wren-CLI (using BigInt) will not be able to complete this task in a reasonable time.

Still takes a while - 11 minutes 20 seconds to get up to base 36 with a limit of 2700.

/* repunit_primes.wren */

import "./gmp" for Mpz
import "./math" for Int
import "./fmt" for Fmt
import "./str" for Str

var limit = 2700
var primes = Int.primeSieve(limit)

for (b in 2..36) {
    var rPrimes = []
    for (p in primes) {
        var s = Mpz.fromStr(Str.repeat("1", p), b)
        if (s.probPrime(15) > 0) rPrimes.add(p)
    }
    Fmt.print("Base $2d: $n", b, rPrimes)
}
Output:
Base  2: [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281]
Base  3: [3, 7, 13, 71, 103, 541, 1091, 1367, 1627]
Base  4: [2]
Base  5: [3, 7, 11, 13, 47, 127, 149, 181, 619, 929]
Base  6: [2, 3, 7, 29, 71, 127, 271, 509, 1049]
Base  7: [5, 13, 131, 149, 1699]
Base  8: [3]
Base  9: []
Base 10: [2, 19, 23, 317, 1031]
Base 11: [17, 19, 73, 139, 907, 1907, 2029]
Base 12: [2, 3, 5, 19, 97, 109, 317, 353, 701]
Base 13: [5, 7, 137, 283, 883, 991, 1021, 1193]
Base 14: [3, 7, 19, 31, 41, 2687]
Base 15: [3, 43, 73, 487, 2579]
Base 16: [2]
Base 17: [3, 5, 7, 11, 47, 71, 419]
Base 18: [2]
Base 19: [19, 31, 47, 59, 61, 107, 337, 1061]
Base 20: [3, 11, 17, 1487]
Base 21: [3, 11, 17, 43, 271]
Base 22: [2, 5, 79, 101, 359, 857]
Base 23: [5]
Base 24: [3, 5, 19, 53, 71, 653, 661]
Base 25: []
Base 26: [7, 43, 347]
Base 27: [3]
Base 28: [2, 5, 17, 457, 1423]
Base 29: [5, 151]
Base 30: [2, 5, 11, 163, 569, 1789]
Base 31: [7, 17, 31]
Base 32: []
Base 33: [3, 197]
Base 34: [13, 1493]
Base 35: [313, 1297]
Base 36: [2]