Explicit primality criteria for $(p-1)p^n-1$

Abstract

Deterministic polynomial time primality criteria for 2 n - 1 have been known since the work of Lucas in 1876-1878. Little is known, however, about the existence of deterministic polynomial time primality tests for numbers of the more general form N n = (p - 1) p n - 1, where p is any fixed prime. When n > (p - 1)/2 we show that it is always possible to produce a Lucas-like deterministic test for the primality of N n which requires that only O(q (p + log q) + p 3 + log N n ) modular multiplications be performed modulo N n , as long as we can find a prime q of the form 1 + kp such that N n k - 1 is not divisible by q. We also show that for all p with 3 < p < 10 7 such a q can be found very readily, and that the most difficult case in which to find a q appears, somewhat surprisingly, to be that for p = 3. Some explanation is provided as to why this case is so difficult.