Mersenne Number and Mersenne Prime Calculator

The Mersenne Number and Mersenne Prime Calculator is a free online tool for checking whether a number is a Mersenne number or a Mersenne prime.

Mersenne Number Calculator

Eg, Is 255 a Mersenne number? Enter 255 into the input box and click Calculate button.

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Definition of Mersenne Number and Mersenne Prime

A Mersenne number is a positive integer that can be represented in the form of 2^p – 1, where p is a prime number, denoted as M_p. According to this definition, we can know that the Mersenne number is an odd number. But the Mersenne number is not necessarily a prime number, it may be a composite number. Such as p = 11, M₁₁ = 2¹¹ – 1 = 2047 = 23 * 89. Obviously, M₁₁ is a composite number.

When the Mersenne number is prime, that is, the value of 2^p – 1 is a prime number, then this Mersenne number is called a Mersenne prime. In other words, when p is prime and 2^p – 1 is also prime, it is called Mersenne prime. From this, we can know the difference between Mersenne numbers and Mersenne primes:

Mersenne primes must be Mersenne numbers, but Mersenne numbers are not necessarily Mersenne primes.

So far, 51 Mersenne primes have been discovered.

For example, p = 2, the Mersenne number is M₂ = 2² – 1 = 3, 3 is a prime number, so it is also a Mersenne prime at this time.

Another example is p = 23, the Mersenne number is M₂₃ = 2²³ – 1 = 8388608 – 1 = 8388607. Using the Prime Factorization Calculator, we can get 8388607 = 47 * 178481, which can be factored. Therefore, M₂₃ is a composite number, which means that M₂₃ is not a Mersenne prime.

First 20 Mersenne numbers

- p = 2, M₂ = 2² – 1 = 3
- p = 3, M₃ = 2³ – 1 = 7
- p = 5, M₅ = 2⁵ – 1 = 31
- p = 7, M₇ = 2⁷ – 1 = 127
- p = 11, M₁₁ = 2¹¹ – 1 = 2047
- p = 13, M₁₃ = 2¹³ – 1 = 8191
- p = 17, M₁₇ = 2¹⁷ – 1 = 131071
- p = 19, M₁₉ = 2¹⁹ – 1 = 524287
- p = 23, M₂₃ = 2²³ – 1 = 8388607
- p = 29, M₂₉ = 2²⁹ – 1 = 536870911
- p = 31, M₃₁ = 2³¹ – 1 = 2147483647
- p = 37, M₃₇ = 2³⁷ – 1 = 137438953471
- p = 41, M₄₁ = 2⁴¹ – 1 = 2199023255551
- p = 43, M₄₃ = 2⁴³ – 1 = 8796093022207
- p = 47, M₄₇ = 2⁴⁷ – 1 = 140737488355327
- p = 53, M₅₃ = 2⁵³ – 1 = 9007199254740991
- p = 59, M₅₉ = 2⁵⁹ – 1 = 576460752303423488
- p = 61, M₆₁ = 2⁶¹ – 1 = 2305843009213693952
- p = 67, M₆₇ = 2⁶⁷ – 1 = 147573952589676412928
- p = 71, M₇₁ = 2⁷¹ – 1 = 2361183241434822606848

First 10 Mersenne primes

3
7
31
127
8191
131071
524287
2147483647
2305843009213693951
618970019642690137449562111

How to use the Mersenne number and Mersenne prime calculator

The procedure to use the Mersenne number and Mersenne prime calculator is as follows:

1Enter the positive number.
2Click calculate button to check whether the input integer is a Mersenne number or a Mersenne prime.
3Click the Reset button to start a new verification.

Solved examples using the Mersenne number and Mersenne prime calculator

Example 1: Is 255 a Mersenne number?

Enter 25 into the input box and click Calculate button, as shown in the figure, 25 is not a Mersenne number.

Example 2: Is 63 a Mersenne prime number?

Enter 63 into the input box and click Calculate button, as shown in the figure, 63 is not a Mersenne prime number.