The Numbers of Thousand Place of Mersenne Primes

Mersenne primes are a special kind of primes, which are an important content in number theory. The study of Mersenne primes becomes one of hot topics of the nowadays science. Searching for Mersenne primes is very challenging in scientific researches. In this paper, the numbers of thousand place of Mersenne primes are studied, and the conclusion is presented by using the Chinese remainder theorem.


Introduction
In 300 BC, ancient Greek mathematician Euclid proved that there are infinitude primes by contradiction, and raised that a small amount of prime numbers could be expressed in numbers of the form , where is a 2 p 1 p prime.Afterward, many famous mathematicians have researched the prime numbers of this special formulation.
In 1644, French mathematician M. Mersenne stated in the preface to his Cogitata Physica-Mathematica that the numbers were primes for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers .Mersenne's (incorrect) conjecture fared only slightly better than Regius', but still got his name attached to these numbers.The formulation of is named as "Mersenne number" and expressed as  As the increase of exponent , searching for Mersenne primes is very challenging.Studying Mersenne primes, not only the advanced theory and practiced skills are needed, but also the arduous calculations are needed to validate whether a Mersenne number is a prime or not [2].The research of Mersenne primes is very abundant in the contemporary theoretical and practical value.The research of Mersenne primes greatly promotes the development of mathematics, especially number theory.In addition, many well-known mathematics problems, such as Goldbach conjecture, twin primes, Riemann conjecture, etc. are inextricably linked to Mersenne primes.Therefore, the research of Mersenne primes can also speed up the solution to those problems [3].And the research of Mersenne primes can promote distributed computing and programming arts.It not only requires well-designed distributed architecture, but also improves the numerical calculation methods and algorithm design arts.[4] There are only 47 known Mersenne primes [5].
p In [6,7], the last number and the ten place number have been studied, and conclusions have been obtained.In this paper, the numbers of thousand place of Mersenne primes are studied, and the conclusion is presented by using the Chinese remainder theorem.
Then, the number of the thousand place of Mersenne primes is 0; If Then, the number of the thousand place of Mersenne primes is 1; If Then, the number of the thousand place of Mersenne primes is 2; If Then, the number of the thousand place of Mersenne primes is 3; Then, the number of the thousand place of Mersenne primes is 7; If Then, the number of the thousand place of Mersenne primes is 8; If Then, the number of the thousand place of Mersenne primes is 9.

Preliminaries
In order to prove the theorem, we need the following lemma.
Lemma (The Chinese Remainder Theorem) Let 1 2 be pair-wise relatively prime positive integers.Then the system of congruencies , , , r m m m  x a m x a m x a m

2 S
. The sequence S(p) is computed modulo to save time.The Lucas-Lehmer test is ideal for binary computers because the division of 2 2 p  1 p  (in binary) can be done by using rotation and addition only.In 1992, Chinese mathematician and linguist Ha-izhong Zhou [1] presented the well-known Zhou conjecture on the distribution of Mersenne primes in the natural number system2 n  Mersenne primes.At the same time he gave the deduction: If , are composites, and then these numbers are not considered.So we can draw the conclusion.That completes the proof.