# Mersenne Sieve

by arydberg
Tags: mersenne, sieve
 P: 96 As a consequence of Theorem 18 from Hardy-Wright, we have the following Corollary: For two natural numbers 1 < a and b: a${^b}$ - 1 is composite if a > 2 (because (a - 1) divides a$^{b}$ - 1); or in the case a = 2: if b = s * t (because 2$^{s}$ - 1 divides 2$^{s*t}$ - 1
 P: 96 I think, my Corollary helps to delete all 2${^b}$ - 1 with a composite b; but unfortunately, there is no help for sieving the Mersenne primes: Theorem 18 by HW says (in short): 'If 2${^b} - 1$ is a prime, then b is a prime'; and the 'other way round' is not valid