Primes p such that there are 4 values of m such that p divides m! + 1.
59, 61, 79, 83, 137, 307, 383, 397, 479, 503, 563, 571, 599, 607, 631, 673, 683, 727, 797, 947, 983, 1019, 1033, 1151, 1637, 1747, 1867, 1871, 1877, 2371, 2377, 2383, 2423, 2591, 2659, 2699, 2707, 2711, 2731, 2851, 2879, 2903, 2969, 3109, 3307, 3343, 3373
1
By Wilson’s therom, p - 1 is always one of the values.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
Eric W. Weisstein, MathWorld: Wilson’s Theorem
(Mma) nn = 4; t = {}; p = 1; While[Length[t] < 50, p = NextPrime[p]; cnt = 0; f = 1; Do[f = Mod[f*m, p]; If[f + 1 == p, cnt++], {m, p - 1}]; If[cnt == nn, AppendTo[t, p]]]; t
nonn
T. D. Noe, Jun 12 2015