Mersenne Number Calculator

This Mersenne number calculator computes 2 to the power p, minus 1, for an exponent you choose. Enter p and the calculator returns the Mersenne number, how many decimal digits it has, and whether p itself is prime, which is a necessary first condition for the Mersenne number to be prime. Mersenne numbers are named after Marin Mersenne and are central to the search for very large primes: the largest known primes are almost all Mersenne primes, found by the distributed GIMPS project. A Mersenne number can only be prime if its exponent is prime, though that alone is not enough. The formula, a worked example and the assumptions are set out below.

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Exponent p (1 to 5000)

Number of digits

Is p prime?

The formula

The Mersenne number for exponent p is M_p = 2^p minus 1. If M_p is prime it is a Mersenne prime, and a Mersenne prime requires p to be prime (but a prime p does not guarantee one).

Worked example

For p = 7, M_7 = 2^7 minus 1 = 128 minus 1 = 127, which is prime, so 127 is a Mersenne prime. For p = 11, M_11 = 2047 = 23 times 89, not prime. Enter 7 to confirm.

Frequently asked questions

What is a Mersenne number?

A number of the form 2 to the power p, minus 1. When it is also prime it is called a Mersenne prime.

Why must p be prime?

If p is composite then 2^p minus 1 factors, so it cannot be prime. A prime p is necessary but not sufficient.

Why are the largest known primes Mersenne primes?

There is a very efficient primality test (Lucas-Lehmer) for Mersenne numbers, which makes searching them practical.

Who this calculator is for

This calculator is for students, programmers and anyone exploring number theory.

What this calculator assumes

You enter a whole number within the stated range.
The exact integer algorithm is used.
Very large results are shown grouped for readability.

Formula and sources

Related calculators

Prime Factorization Calculator: break a number into prime factors.
LCM Calculator: lowest common multiple.
GCF Calculator: greatest common factor.

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