Wieferich primes

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In number theory, a Wieferich prime is a prime number p such that p² evenly divides 2^{(p − 1)} − 1 .

It is conjectured that there are infinitely many Wieferich primes, but as of March 2021,only two have been identified.

Task

Write a routine (function procedure, whatever) to find Wieferich primes.

Use that routine to identify and display all of the Wieferich primes less than 5000.

See also

OEIS A001220 - Wieferich primes

C++

<lang cpp>#include <cstdint>

include <iostream>
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;

}

uint64_t modpow(uint64_t base, uint64_t exp, uint64_t mod) {

   if (mod == 1)
       return 0;
   uint64_t result = 1;
   base %= mod;
   for (; exp > 0; exp >>= 1) {
       if ((exp & 1) == 1)
           result = (result * base) % mod;
       base = (base * base) % mod;
   }
   return result;

}

std::vector<uint64_t> wieferich_primes(uint64_t limit) {

   std::vector<uint64_t> result;
   std::vector<bool> sieve(prime_sieve(limit));
   for (uint64_t p = 2; p < limit; ++p)
       if (sieve[p] && modpow(2, p - 1, p * p) == 1)
           result.push_back(p);
   return result;

}

int main() {

   const uint64_t limit = 5000;
   std::cout << "Wieferich primes less than " << limit << ":\n";
   for (uint64_t p : wieferich_primes(limit))
       std::cout << p << '\n';

}</lang>

Output:

Wieferich primes less than 5000:
1093
3511

Factor

Works with: Factor version 0.99 2021-02-05

<lang factor>USING: io kernel math math.functions math.primes prettyprint sequences ;

"Weiferich primes less than 5000:" print 5000 primes-upto [ [ 1 - 2^ 1 - ] [ sq divisor? ] bi ] filter .</lang>

Output:

Weiferich primes less than 5000:
V{ 1093 3511 }

Julia

<lang julia>using Primes

println(filter(p -> (big"2"^(p - 1) - 1) % p^2 == 0, primes(5000))) # [1093, 3511] </lang>

Phix

include mpfr.e
mpz z = mpz_init()
function weiferich(integer p)
    mpz_set_str(z,repeat('1',p-1),2)
    return mpz_fdiv_q_ui(z,z,p*p)=0
end function
printf(1,"Weiferich primes less than 5000: %V\n",{filter(get_primes_le(5000),weiferich)})

Output:

Weiferich primes less than 5000: {1093,3511}

Raku

<lang perl6>put "Weiferich primes less than 5000: ", join ', ', ^5000 .grep: { .is-prime and not ( exp($_-1, 2) - 1 ) % .² };</lang>

Output:

Weiferich primes less than 5000: 1093, 3511

Wren

Library: Wren-math

Library: Wren-big

<lang ecmascript>import "/math" for Int import "/big" for BigInt

var primes = Int.primeSieve(5000) System.print("Weiferich primes < 5000:") for (p in primes) {

   var num = (BigInt.one << (p - 1)) - 1
   var den = p * p
   if (num % den == 0) System.print(p)

}</lang>

Output:

Weiferich primes < 5000:
1093
3511