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Mersenne primes and perfect numbers are tw … Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1. The numbers p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89. Meanwhile, perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2p − 1 × (2p − 1), where 2p − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form. For instance, in the case of p = 2, 22 − 1 = 3 is prime, and 22 − 1 × (22 − 1) = 2 × 3 = 6 is perfect. It is currently an open problem as to whether there are an infinite number of Mersenne primes and even perfect numbers. The frequency of Mersenne primes is the subject of the Lenstra–Pomerance–Wagstaff conjecture, which states that the expected number of Mersenne primes less than some given x is (eγ / log 2) × log log x, where e is Euler's number, γ is Euler's constant, and log is the natural logarithm. It is also not known if any odd perfect numbers exist; various conditions on possible odd perfect numbers have been proven, including a lower bound of 101500. The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2022, there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 of which have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS. New Mersenne primes are found using the Lucas-Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers. The displayed ranks are among indices currently known as of 2022; while unlikely, ranks may change if smaller ones are discovered. According to GIMPS, all possibilities less than the 48th working exponent p = 57,885,161 have been checked and verified as of October 2021. The discovery year and discoverer are of the Mersenne prime, since the perfect number immediately follows by the Euclid–Euler theorem. Discoverers denoted as "GIMPS / name" refer to GIMPS discoveries with hardware used by that person. Later entries are extremely long, so only the first and last 6 digits of each number are shown.nd last 6 digits of each number are shown.
, 梅森素数与完全数是数论里关系密切的自然数。梅森素数以数学家、神学家、修士马兰·梅森命 … 梅森素数与完全数是数论里关系密切的自然数。梅森素数以数学家、神学家、修士马兰·梅森命名，是能以2n－1表示、且n为正整数的质数，如梅森素数3就能写成22－1。梅森素数在上述表达式对应的数n一定是质数，但n是质数不代表得出的结果就是梅森素数，如211－1＝2047＝23×89。完全数是等于真因数之和的自然数，真因数即自然数除自身外的因数。例如6就是完全数，因数分别是1、2、3、6且1＋2＋3＝6。 根据欧几里得部分证明、萊昂哈德·歐拉完全证明的歐幾里得-歐拉定理可知梅森素数与已知完全数一一对应：只有能换算成公式2n－1×（2n－1），且2n − 1是梅森素数的偶数是完全数。以n＝2为例，22－1＝3为质数，22－1×（22－1）＝2×3＝6为完全数。 梅森素数与完全数是否无穷尽目前还是未解决的数学问题，伦斯特拉－波默朗斯－瓦格斯塔夫猜想的主题便是梅森素数频率，推断比x小的梅森素数期望个数為（eγ/log2）×log log x，其中e是欧拉数，γ是欧拉常数，log是自然對數。已经发现的完全数都是偶数，但尚未排除存在奇数完全数的可能。已證明奇完全数必滿足某些條件，如不小於101500。 下表列出所有已知梅森素数、完全数及对应指数n。截至2021年10月人类共发现51个梅森素数（故也有51个完全数），其中最大的17个均由互联网梅森素数大搜索分布式计算项目发现。新梅森素数是用卢卡斯-莱默检验法发现，这种梅森素数素性测试可用于二进制计算机。 数字按从小到大排列，如果新发现比现有结果小的梅森素数则插入中间。序号后面的问号说明尚待验证。截至2021年10月，互联网梅森素数大搜索已经计算至n＝58,204,879，即第48个梅森素数以前的所有自然数均已验证。发现时间和发现人均指梅森素数，完全数按歐幾里得-歐拉定理计算。发现人列为“互联网梅森素数大搜索：姓名”说明此人拥有的设备采用互联网梅森素数大搜索找到该数。除前八个数不超过十位外，后面的结果都非常长，最长的已有数千万位，故下表仅列出前后各六位数，中间以省略号表示。后面的结果都非常长，最长的已有数千万位，故下表仅列出前后各六位数，中间以省略号表示。
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梅森素数与完全数是数论里关系密切的自然数。梅森素数以数学家、神学家、修士马兰·梅森命 … 梅森素数与完全数是数论里关系密切的自然数。梅森素数以数学家、神学家、修士马兰·梅森命名，是能以2n－1表示、且n为正整数的质数，如梅森素数3就能写成22－1。梅森素数在上述表达式对应的数n一定是质数，但n是质数不代表得出的结果就是梅森素数，如211－1＝2047＝23×89。完全数是等于真因数之和的自然数，真因数即自然数除自身外的因数。例如6就是完全数，因数分别是1、2、3、6且1＋2＋3＝6。 根据欧几里得部分证明、萊昂哈德·歐拉完全证明的歐幾里得-歐拉定理可知梅森素数与已知完全数一一对应：只有能换算成公式2n－1×（2n－1），且2n − 1是梅森素数的偶数是完全数。以n＝2为例，22－1＝3为质数，22－1×（22－1）＝2×3＝6为完全数。 梅森素数与完全数是否无穷尽目前还是未解决的数学问题，伦斯特拉－波默朗斯－瓦格斯塔夫猜想的主题便是梅森素数频率，推断比x小的梅森素数期望个数為（eγ/log2）×log log x，其中e是欧拉数，γ是欧拉常数，log是自然對數。已经发现的完全数都是偶数，但尚未排除存在奇数完全数的可能。已證明奇完全数必滿足某些條件，如不小於101500。但尚未排除存在奇数完全数的可能。已證明奇完全数必滿足某些條件，如不小於101500。
, Mersenne primes and perfect numbers are tw … Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1. The numbers p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89. Meanwhile, perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6.s of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6.
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梅森素数与完全数集合
, List of Mersenne primes and perfect numbers
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