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Mersenne prime - перевод на русский
![The [[Gaussian prime]]s with norm less than 500](https://commons.wikimedia.org/wiki/Special:FilePath/Gaussian primes.png?width=200 "The [[Gaussian prime]]s with norm less than 500")
![relative error]] of <math>\tfrac{n}{\log n}</math> and the logarithmic integral <math>\operatorname{Li}(n)</math> as approximations to the [[prime-counting function]]. Both relative errors decrease to zero as <math>n</math> grows, but the convergence to zero is much more rapid for the logarithmic integral.](https://commons.wikimedia.org/wiki/Special:FilePath/Prime-counting relative error.svg?width=200 "relative error]] of <math>\tfrac{n}{\log n}</math> and the logarithmic integral <math>\operatorname{Li}(n)</math> as approximations to the [[prime-counting function]]. Both relative errors decrease to zero as <math>n</math> grows, but the convergence to zero is much more rapid for the logarithmic integral.")
общая лексика
простое число
целое число, которое делится без остатка только на себя и на 1. Для поиска небольших простых чисел широко используется алгоритм "Решето Эратосфена"
Смотрите также
математика
нечетное простое число
Определение
Википедия
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p.
The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in the OEIS).
Numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89.
Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality.
As of October 2022, 51 Mersenne primes are known. The largest known prime number, 282,589,933 − 1, is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.
