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
Factor
<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
is_weiferich(p) = (big"2"^(p - 1) - 1) % p^2 == 0
function weiferich_to(N)
n = 0 while (n = nextprime(n + 1)) < N is_weiferich(n) && print(n, " ") end
end
weiferich_to(5000) # prints 1093 3511 </lang>
Phix
include mpfr.e function weiferich(integer p) mpz p2pm1m1 = mpz_init() mpz_ui_pow_ui(p2pm1m1,2,p-1) mpz_sub_ui(p2pm1m1,p2pm1m1,1) return mpz_fdiv_q_ui(p2pm1m1,p2pm1m1,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
<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