Mersenne prime

A prime number of the form 2^p − 1, where p is prime. Every Mersenne prime has a prime exponent, but a prime exponent does not guarantee a Mersenne prime.

Lucas–Lehmer test: A deterministic test for whether 2^p − 1 is prime when p is an odd prime. It makes testing enormous Mersenne numbers practical using a short recurrence.
Marin Mersenne: A French Minim friar and scholar who studied numbers and corresponded with European intellectuals. His 1644 work listed conjectured prime exponents for numbers of this form.
Euclid–Euler theorem: The theorem that even perfect numbers correspond exactly to primes of the form 2^p − 1. It turns each Mersenne prime into an even perfect number, and classifies all such numbers.
Mersenne prime 2^31 − 1: The prime 2,147,483,647, equal to 2^31 − 1. It is a familiar Mersenne prime and the largest signed 32-bit integer.
Mersenne number: An integer of the form 2^n − 1, where n is a positive integer. Every Mersenne prime is a Mersenne number, though most Mersenne numbers are composite.
Euclid: An ancient Greek mathematician whose surviving work includes foundational results in number theory and geometry. Euclid proved that 2^p − 1 prime produces an even perfect number.
Perfect number: A positive integer equal to the sum of its proper positive divisors. The formula 2^(p−1)(2^p−1) constructs one whenever 2^p−1 is prime.
Mersenne prime 2^127 − 1: The 39-digit prime 2^127 − 1, discovered by Édouard Lucas in 1876. Its proof predates electronic computers and was a major nineteenth-century result.
Mersenne conjecture: The unproved claim that infinitely many Mersenne primes exist. The scarcity of known examples leaves their infinitude unresolved.
Leonhard Euler: An eighteenth-century mathematician who made major contributions across mathematics and science. Euler proved that every even perfect number has Euclid’s Mersenne-prime form.

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