Mersenne Prime Checker
Test whether 2ⁿ − 1 is a Mersenne prime for a given exponent n.
Inputs
Exponent for 2ⁿ − 1. Limited to 61 for exact integer computation.
Is Mersenne Prime?
Yes
2ⁿ − 1
127
Is Exponent Prime?
Yes
Step by step
Mersenne number
M(7) = 2^7 − 1 = 127
Exponent prime?
= Yes (required)
M(n) prime?
= Yes — Mersenne prime!
How it works
A Mersenne prime is a prime number of the form Mₙ = 2ⁿ − 1. A necessary (but not sufficient) condition is that n itself is prime. Known Mersenne prime exponents include 2, 3, 5, 7, 13, 17, 19, 31, 61. This calculator is limited to n ≤ 61 because trial-division primality testing of the resulting number (up to 2⁶¹−1 ≈ 2.3×10¹⁸) is the largest feasible in-browser.
Formula
Mersenne number
M(n) = 2^n − 1
- n
- Exponent (must be prime for M(n) to possibly be prime)
Frequently Asked Questions
Why must the exponent be prime?
If n = a × b with a,b > 1, then 2ⁿ − 1 is divisible by 2ᵃ − 1, so it cannot be prime. Therefore n must be prime — but that alone is not sufficient (e.g. M(11) = 2047 = 23 × 89).
What are the known Mersenne primes?
As of 2024, 51 Mersenne primes are known. The smallest exponents are 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, … The largest known prime is always a Mersenne prime.