关于广义Mersenne 数的素因数

journal6 ›› 2000, Vol. 21 ›› Issue (1): 32-33.

关于广义Mersenne 数的素因数

(茂名教育学院数学系, 广东茂名 525000)

出版日期:2000-03-15 发布日期:2013-01-16
作者简介:陈荣基( 1965~ ) ,男, 广东省高州市人,茂名教育学院数学系讲师,主要从事数论研究.

On the Prime Divisors of Generalized Mersenne Numbers

( Dept. of Math. ,Maoming Educational College ,Maoming 525000, Guangdong, China)

Online:2000-03-15 Published:2013-01-16

摘要/Abstract

摘要：设p是奇素数, a 是大于1的正整数,又设 X ( a, p ) = ( a^p- 1) / ( a- 1) , Y( a, p ) = ( a^p+ 1) / ( a+ 1) ,当 q= 2p+1 是素数时,如果( a/ q )= 1且 qa- 1,则 q 必为X( a, p )的素因数; 如果( a/ q )= - 1 且 qa + 1, 则 q 必为 Y( a, p )的素因数,其中( a/ q)是 Legendre 符号.

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

陈荣基. 关于广义Mersenne 数的素因数[J]. journal6, 2000, 21(1): 32-33.

CHEN Rong-Ji. On the Prime Divisors of Generalized Mersenne Numbers[J]. journal6, 2000, 21(1): 32-33.