梅森素数与牛顿迭代

梅森素数是数论研究的一项重要内容，也是当今科学探索的热点和难点之一。卢卡斯定理是判别梅森数是否为素数的第一个重要定理，卢卡斯-雷默测试是在卢卡斯定理基础上改进后的现在已知的检验梅森数素性的最好方法。牛顿迭代法可以用来求平方根√n的近似值。本文首先揭示了卢卡斯定理与√5的牛顿迭代之间的惊人联系，然后揭示了卢卡斯-雷默测试与√3的牛顿迭代之间的惊人联系，继而揭示了梅森素数的一个同余性质与√4的牛顿迭代之间的惊人联系，又通过√2的牛顿选代得出了梅森素数的一个新的同余性质，并猜测由该性质产生的数列具有与斐波那契数列相类似的漂亮性质，接着通过√6的牛顿迭代提出了p为4k＋1形素数时梅森数Mp为素数所应满足的充要条件的猜想，最后提出了基于梅森素数同余性质的梅森数素性检验新方法的猜想。

Mersenne Prime and Newton＇s Iteration

The Mersenne prime is the important content of number theory research. It is also one of today＇ s hot and difficult scientific explorations. Lucas theorem is the first important tool for determining the primality of Mersennc numbers. So far Lucas-Lehmer test is the fastest and most effective tool for determining the primality of Mcrsenne numbers. Newton＇s iteration is an algorithm for computing the square root √n of a number n. This article first reveals the amazing relationship between Lucas theorem and Newton＇s iteration of √5, then reveals the amazing relationship between Lucas-Lehmer test and Newton＇s iteration of √3 , and then reveals the amazing relationship between a divisibility property of Mersenne primes and Newton＇s iteration of √4, and also proves a new divisibility property of Mersenne primes by Newton＇s iteration of √2, and guesses the sequence derived from the new divisibility property has the same wonderful property as Pibonacci sequence, and then guesses that Mp should meet necessary and sufficient conditions if Mp is also a prime when p is a prime of the form 4k＋l by Newton＇s iteration of √6 , finally makes a conjecture about the new primality test for Mersenne numbers by the congruence property of Mersenne primes.