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41 is: the 13th smallest prime number. The next is 43, making both twin primes. the sum of the first six prime numbers (2 + 3 + 5 + 7 + 11 + 13). the 12th supersingular prime [1] a Newman–Shanks–Williams prime. [2] the smallest Sophie Germain prime to start a Cunningham chain of the first kind of three terms, {41, 83, 167}.
A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. ... 8n+1: 17, 41, 73, 89, 97, 113, ...
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 2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1.
Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]
The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.
A number is only considered a Mersenne prime if it can be written in the form 2ᵖ-1. Unlike other large prime numbers used in some applications to protect internet security, Mersenne primes are ...
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (17, 19) or (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or ...
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