Mersenne Prime — Definition, Formula & Examples

A Mersenne prime is a prime number that can be written in the form

2^p - 1

, where

p

itself is a prime number. For example,

2^3 - 1 = 7

is a Mersenne prime, while

2^{11} - 1 = 2047 = 23 \times 89

is not.

A Mersenne prime is a prime number $M_p = 2^p - 1$ for some prime $p$ . Not every prime exponent $p$ yields a prime; $M_p$ is called a Mersenne number regardless of primality, and a Mersenne prime only when $M_p$ is itself prime.

Key Formula

M_p = 2^p - 1

Where:

$M_p$ = The Mersenne number associated with exponent p
$p$ = A prime number used as the exponent

How It Works

To check whether a number is a Mersenne prime, first confirm the exponent

p

is prime — if

p

is composite, then

2^p - 1

is guaranteed to be composite. Then test whether

2^p - 1

is prime. For small exponents you can factor directly, but for large ones, the Lucas–Lehmer test provides an efficient method. The Great Internet Mersenne Prime Search (GIMPS) has used this test to discover the largest known primes, all of which are Mersenne primes.

Worked Example

Problem: Determine which of the exponents p = 2, 3, 5, 7, 11 produce Mersenne primes.

p = 2: Compute the Mersenne number and check primality.

M_2 = 2^2 - 1 = 3 \quad \text{(prime ✓)}

p = 3: Compute and check.

M_3 = 2^3 - 1 = 7 \quad \text{(prime ✓)}

p = 5: Compute and check.

M_5 = 2^5 - 1 = 31 \quad \text{(prime ✓)}

p = 7: Compute and check.

M_7 = 2^7 - 1 = 127 \quad \text{(prime ✓)}

p = 11: Compute and factor.

M_{11} = 2^{11} - 1 = 2047 = 23 \times 89 \quad \text{(not prime ✗)}

Answer: The exponents 2, 3, 5, and 7 produce Mersenne primes (3, 7, 31, 127). The exponent 11 does not, since 2047 is composite.

Why It Matters

Mersenne primes are central to the search for the largest known prime numbers — as of 2024, every record-holding prime is a Mersenne prime. They also connect to perfect numbers: every even perfect number has the form

2^{p-1}(2^p - 1)

where

2^p - 1

is a Mersenne prime. This link makes them a key topic in any number theory course.

Common Mistakes

Mistake: Assuming every prime exponent p produces a Mersenne prime.

Correction: A prime exponent is necessary but not sufficient. For instance, p = 11 is prime, yet 2^11 − 1 = 2047 = 23 × 89 is composite. You must verify the primality of 2^p − 1 separately.