Double Mersenne Prime

Last Updated : 23 Jul, 2025

A Double Mersenne prime is a special type of prime number derived from another prime called a Mersenne prime. A Mersenne prime is a number like 2n - 1, where n is a prime number.

To get a Double Mersenne prime, you take a Mersenne prime and use it as the exponent in the formula M_{M_{n}} = 2^{2^{n-1}} - 1. Only a few of these numbers are known, which makes them limited and interesting!

Note: We only have found 4 double Mersenne prime as of October 2024.

Formula of Double Mersenne Prime:

M_{M_{n}} = 2^{n-1} - 1

List of Double Mersenne Prime

For n

Mersenne Prime Mn=(2n-1)

Double Mersenne PrimeM_{M_{n}} = 2^{2^{n-1}}-1

2

M2 = 22 - 1 = 3

M_{M_{2}} = 23 - 1 = 7

3

M3 = 23 - 1 = 7

M_{M_{3}} = 27 - 1 = 127

5

M5 = 25 -1 = 31

M_{M_{5}} = 231 - 1 = 2,147,483,647

7

M7 = 27 - 1 = 127

M_{M_{7}} = 2127 - 1 = 170,141,183,460,469,231,731,687,303,715,884,105,727

The first four Double Mersenne numbers are known to be prime, but that is not the case for the next four ((M_{M_{13}},\ M_{M_{17}},\ M_{M_{19}}, \ M_{M_{31}}) have not been confirmed as primes and are considered composite until proven otherwise.

Conclusion

Double Mersenne Prime are very limited and interesting prime numbers derived from the Mersenne primes using it as exponent. The value of Double Mersenne primes grows rapidly; for example, the third double Mersenne prime has 10 digits while the fourth has 39 digits. The research for the Double Mersenne prime number continues but we only found 4 as of October 2024.

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