Primality test for numbers M with a large power of 5 dividing M4−1

AbstractThe Quintic Reciprocity Law is used to produce an algorithm, that runs in polynomial time, and that determines the primality of numbers M, such that M4−1, is divisible by a power of 5 which is larger that M, provided that a small prime p, p≡1(mod5) is given, such that M, is not a fifth power modulo p. The same test equations are used for all such M.A sufficiency test, together with its probability of succeeding in determining primality is given when the condition on M modulo p is omitted