关键词： 广义Mersenne数, Germain素数, 素因数

Abstract: Let p be an odd prime, and let a be a positive inte ger w th a> 1.Further let X ( a , p ) = ( a^p- 1) / ( a - 1)and Y( a, p ) = ( a^p+ 1) / ( a+ 1) . In this paper we prove that if  q= 2p+ 1 is a prime, then q is a prime divisor of X  ( a, p ) or Y( a, p ) according as ( a/ q) = 1 and q a- 1 or ( a / q) = - 1 and q a+ 1,where ( a/ q ) is the Legendre symbol.

Key words: generalized Mersenne number, Germain prime, prime divisor