Mersenne Primes

Mersenne Primes
Basic Info
Discovered by	Marin Mersenne (September 8, 1588-September 1, 1648)
Number of	Conjectured infinite
Description	A prime which is one less than a power of two
First Few	3, 7, 31, 127, 8,191 , 131,071

Mersenne Primes are numbers that can be expressed in the form  2^p − 1, where p is a prime number. Not all numbers of the form 2^p − 1 are prime, but those which are prime are known as Mersenne primes, named after French mathematician, Marin Mersenne. Numbers of the form 2ⁿ − 1 where n is composite cannot be prime.

Since 1992, when 2^756,839 − 1 was proved prime, the largest known prime number has always been a Mersenne Prime. In 2014,  2^57,885,161 − 1 was found to be prime, which contains 17,425,170 digits. As of 2022, the largest known prime is 2^82,589,933 -1 which has 24,862,048 digits.

Mersenne Primes are closely linked with Perfect numbers (numbers which are the sum of their proper divisors). For any Mersenne Prime, 2^p - 1, the composite number (2^p-1)x2^p-1 is perfect.

The First Few Mersenne Prime Numbers

2² - 1

2³ - 1

2⁵ - 1

2⁷ - 1

2¹³-1

2¹⁷ - 1

2¹⁹ - 1

2³¹ - 1

See also

Double Mersenne Prime