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A152217 Primes p == 1 (mod 3) such that ((p-1)/3)! == 1 (mod p). 0

%I #4 May 29 2022 19:32:09

%S 3571,4219,13669,25117,55897,89269,102121,170647,231019,246247,251431

%N Primes p == 1 (mod 3) such that ((p-1)/3)! == 1 (mod p).

%C The Wilson theorem states that p is prime if and only if (p-1)! = -1 (mod p). If p = 3 (mod 4) then ((p-1)/2)! = +/- 1 (mod p).

%D J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.2.

%H J. B. Cosgrave, <a href="http://staff.spd.dcu.ie/johnbcos/jacobi.htm">Jacobi</a> [From Francois Brunault (brunault(AT)gmail.com), Nov 29 2008]

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Wilson&#39;s_theorem">Wilson's theorem</a>

%e For n = 1 the prime a(1) = 3571 divides 1190! - 1.

%o (PARI) forprime(p=2,30000,if(p%3==1 & ((p-1)/3)!%p==1,print(p)))

%Y Seems to be a subsequence of A002407 and therefore of A003215 (differences of consecutive cubes). See also A058302 and A055939 for the sequences corresponding to ((p-1)/2)! = +/- 1 (mod p).

%K nonn,more

%O 1,1

%A Francois Brunault (brunault(AT)gmail.com), Nov 29 2008, Nov 30 2008

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)