梅森素数的一些注记

 梅森素数历来是数论研究的重要内容，也是当今科学探索的热点和难点之一；而卢卡斯-雷默测试是迄今为止判断梅森数素性最快最有效的工具；周氏猜测是关于梅森素数分布的著名难题。本文首先介绍与梅森素数研究有关的3个重要问题：然后通过对卢卡斯-雷默测试递归数列的研究，揭示了其衍生数列的一个特殊性质，提出相关的猜想；得出卢卡斯-雷默测试的一个关联等式，由该等式与周氏猜测的密切关系，提出相关的猜想；提出了广义卢卡斯-雷默测试的存在性问题，并提出了相关的猜想。结果表明，采用不同的方法对解决梅森素数的有关问题会有所启发和帮助。

Some Notes on Mersenne Primes

A Mersenne prime is a Mersenne number that is a prime. It is known that if 2p-1 is a prime, p will also be a prime, therefore, it makes no difference which definition of Mersenne number is used. The Mersenne prime has long fascinated both amateur and professional mathematicians. It is always an important topic in the number theory and also one of today's hot and difficult scientific problems. Mersenne primes, as usually the largest known primes, are both rare and beautiful. As of January 2010, only 47 Mersenne primes are known. Lucas-Lehmer test is the fastest and most effective tool for determining the primality of Mersenne numbers. Zhou conjecture is the well-known problem of the distribution of Mersenne primes. This paper first discusses three important problems related with Mersenne primes. Then from the recursive sequence of Lucas-Lehmer test, a special nature of the derivative series is shown, a related guess is made; a correlation equation of Lucas-Lehmer test is obtained, the relevant guess is made from the type of the close relationship between the speculation and Zhou conjecture; and the problem of existence of the generalized Lucas-Lehmer test is formulated, together with a related guess. The results show a good prospect in the studies of Mersenne primes.