Mersenne prime

Ancient Greek mathematician Euclid has proved that there are infinite primes , And some prime numbers can be written as “2P-1”（ And the index P And prime numbers ） In the form of , among 17 French mathematician in the th century 、 Marin, the founder of Francis College of Sciences · Mason （Martin Mersenne） Is one of the more outstanding achievements , So the mathematical world will “2P-1” The prime of a type is called “ Mason prime number ”.

1772 year , Euler is blind , It is proved by mental calculation that 231-1（ namely 2147483647） It's No 8 A mason prime number , This record has not been broken for more than 100 years . The following is the process of Euler proving that there are infinite primes , But no one can prove whether there are infinite numbers of Mason primes .

Suppose prime p1,p2,p3……pn There are only so many , Now there are new numbers p=p1*p2*p3*……pn + 1, so p Can't be p1,p2,p3……pn Any integer division , therefore p Prime number , Then it's only n A contradiction , So there are infinite primes .

1996 year , George · Waterman compiled a program for calculating Mason primes , Later it developed into GIMPS（Great Internet Mersenne Prime Search, You can find the latest number from above 48 Information found by Mason prime numbers ） project , After the supercomputer tried , Hope to use the power of the Internet and distributed computing , Looking for a larger Mersenne prime （ Now there are more than 180 Nearly onemillion computers from countries participate in the calculation ）.

In the process of finding Mason prime numbers , It's not aimless . It's easy to prove , If 2P-1 Prime number , be P It must be prime （ So we can first list one “ List of suspicious Mason primes ”, Further calculation principles can be found in here find ）, Prove the following ：

hypothesis 2P-1 In the case of prime numbers ,p But it's a composite , So make p=r*s,r and s Are greater than 1 The positive integer , that xrs-1 It can be disassembled into xs-1 multiply xs(r-1) + xs(r-2) + … + xs + 1. therefore , If p If it is a composite number ,2p-1 It will also be a composite （ Because it can be removed 2s-1 Factor of ）, This is inconsistent with the hypothetical proposition , therefore p It can only be prime . It is worth noting that , about n>1, because x-1 Divisibility xn-1 , So if you want to xn-1 If it is a prime number ,x-1 It has to be equal to 1 了 , therefore x It can only be 2 了 . that , We can get the following inference ： If a and n Are greater than 1 The positive integer , If an-1 If it's a prime number , that a It can only be 2, and n Must be prime .

Professor of mathematics, Central University of Missouri Curtis Cooper The research team led a week ago 1 month 25 The largest known Mersenne prime number was found on the th ——257885161−1（ namely 2 Of 57885161 The power minus 1）; The prime number has 17425170 position , If you print it continuously with ordinary font size , Its length can exceed 65 km .

The distribution of Mason primes is extremely irregular , Even the time distribution of finding Mason prime numbers is extremely irregular , Sometimes I can't find one for many years , And sometimes you find several at a time , It seems more difficult to explore the distribution law of Mason prime numbers than to find new Mason prime numbers . Chinese amateur mathematician Zhou Haizhong is 1992 In, he gave the exact expression of the distribution of Mason primes （“ Zhou Shi guessed ”）：

For prime numbers p And Mason prime Mp=2P-1 , When 22^n<p<22^(n+1) when ,Mp Yes 2n+1-1 individual ; inference ： When p<22^(n+1) when ,Mp Yes 2n+2－n－2 individual .

Mason prime is a powerful tool for testing computer speed , In fact, the process of finding Mason prime numbers also promotes the development of Distributed Computing （ This is often the case when basic disciplines such as mathematics are looking for the current practical significance , But no one can predict how significant it will be for the future development of engineering discipline ）, In the practical field , Mason primes can also be used to encrypt data . Because it's easy to multiply two very large numbers , But if you want to decompose a very large number , It will be very difficult , In this encryption design , Use large prime numbers , The larger the prime number , In theory, the harder it is to be decoded .

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