Primes p such that there are 3 values of m such that p divides m! + 1.
23, 67, 103, 251, 257, 269, 271, 317, 379, 389, 419, 431, 463, 467, 521, 569, 601, 613, 619, 647, 691, 709, 739, 787, 809, 823, 829, 881, 883, 967, 991, 1021, 1031, 1091, 1117, 1129, 1201, 1213, 1319, 1321, 1367, 1459, 1483, 1499, 1531, 1559, 1609, 1657, 1723
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 = 3; 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