Quintic reciprocity and primality test for numbers of the form M=A5n ± ωn

The Quintic Reciprocity Law is used to produce an algorithm, that runs in polynomial time, and that determines the primality of numbers M such that M-4 - 1 is divisible by a power of 5 which is larger that rootM, provided that a small prime p, p = 1(mod5) is given, such that nl is not a fifth power module p. The same test equations are used for all such M. If M is a fifth power module p, a sufficient condition that determines the primality of M is given.