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Home Page > Math > Basic Math Operations

Mersenne Prime Checker

Test a candidate of the form 2^p − 1 with MiniWebtool’s Mersenne Prime Checker. A Lucas-Lehmer iteration trace explains the primality test, with binary-pattern visuals and the related perfect number for qualifying primes.

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Lucas–Lehmer: M_p is prime exactly when s → s² − 2 ends at 0

Enter an exponent p below to test whether 2^p − 1 is a Mersenne prime, with the Lucas–Lehmer trace and the bit pattern.

Pick a famous exponent to test — each one runs in milliseconds:

✦ Known prime \(M_p\) p = 13 p = 17 p = 31 p = 61 p = 127
✕ Composite \(M_p\) p = 11 p = 23 p = 37 p = 67
⚡ Big ones p = 521 p = 1279 p = 2281 p = 4253
2^

Any positive integer from 1 to 5,000. For larger exponents use specialized software like Prime95.

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About Mersenne Prime Checker

Welcome to the Mersenne Prime Checker, an interactive tool that tests whether \(2^p - 1\) is a Mersenne prime for any exponent \(p\) up to 5000. The tool runs the celebrated Lucas-Lehmer primality test, shows an animated iteration trace of the recurrence \(S_i = S_{i-1}^2 - 2 \pmod{M_p}\), visualizes the binary bit pattern (a defining signature of every Mersenne number), and — when the result is prime — pairs it with the corresponding even perfect number via the Euclid-Euler theorem.

What Is a Mersenne Prime?

A Mersenne number is a number of the form \(M_p = 2^p - 1\). When \(M_p\) is itself prime, it is called a Mersenne prime. The name honors Marin Mersenne (1588-1648), the French monk who catalogued the early cases and conjectured which exponents up to 257 yielded primes — a list that turned out to be partly wrong, but launched three centuries of research.

Mersenne Prime
$$M_p = 2^p - 1 \;\; \text{is prime, where } p \text{ itself must be prime}$$

The first few Mersenne primes, in order:

As of 2024, exactly 52 Mersenne primes are known. The current record is \(M_{136{,}279{,}841}\), discovered in October 2024 by the GIMPS distributed-computing project — a number with 41,024,320 decimal digits.

The Lucas-Lehmer Test

The reason Mersenne primes dominate the record books is a specialized, extremely fast primality test discovered by Édouard Lucas (1878) and simplified by Derrick Lehmer (1930):

Lucas-Lehmer Test
$$S_0 = 4, \quad S_i = S_{i-1}^2 - 2 \pmod{M_p}$$

For prime \(p \geq 3\): \(\;M_p\) is prime \(\iff S_{p-2} \equiv 0 \pmod{M_p}\)

The test requires only \(p-2\) modular squarings — roughly \(O(p^3)\) bit operations with schoolbook multiplication, or \(O(p^2 \log p \log\log p)\) with FFT. Compare this with general-purpose primality tests on numbers the size of \(M_p\) (millions of digits), which would be completely infeasible. The Lucas-Lehmer shortcut is what makes the Mersenne-prime search possible.

Why Must \(p\) Be Prime?

If \(p = a \cdot b\) with \(a, b > 1\), a classical identity shows that \(2^a - 1\) divides \(2^{ab} - 1\):

Factoring identity
$$2^{ab} - 1 = (2^a - 1)\left(2^{a(b-1)} + 2^{a(b-2)} + \cdots + 2^a + 1\right)$$

So if the exponent is composite, \(M_p\) is automatically composite. The converse is false: \(p\) being prime does not guarantee \(M_p\) is prime. For example, \(p = 11\) is prime but \(M_{11} = 2047 = 23 \times 89\).

Mersenne Primes and Perfect Numbers (Euclid-Euler)

Euclid observed around 300 BC that if \(2^p - 1\) is prime, then \(2^{p-1}(2^p - 1)\) is a perfect number — a number equal to the sum of its proper divisors. Euler later proved the converse: every even perfect number arises this way.

Euclid-Euler Theorem
$$N \text{ is an even perfect number} \iff N = 2^{p-1}(2^p - 1),\;\; 2^p - 1 \text{ prime}$$

So finding a new Mersenne prime instantly produces a new perfect number. The first four even perfect numbers are 6, 28, 496, and 8128 — known since antiquity. Whether any odd perfect number exists remains an unsolved problem more than 2,300 years old.

The Binary Bit Pattern

Every Mersenne number has a uniquely clean binary representation: \(2^p\) in binary is \(1\) followed by \(p\) zeros, so \(2^p - 1\) is exactly \(p\) consecutive 1-bits:

M_5 = 2^5 − 1 = 111112 = 31
M_7 = 2^7 − 1 = 11111112 = 127

This is why the tool visualizes each bit as its own tile — the bit pattern is the visual signature of a Mersenne number, independent of whether the number is prime.

How to Use This Calculator

  1. Enter an exponent \(p\): any positive integer from 1 to 5,000.
  2. Click Check: the tool first checks whether \(p\) is prime; if not, it explains why \(M_p\) must be composite.
  3. For prime \(p\): the Lucas-Lehmer recurrence runs \(p - 2\) iterations modulo \(M_p\).
  4. Explore the output: verdict banner, 6-row iteration trace (with "..." for omitted middle steps on large \(p\)), decimal and binary forms of \(M_p\), and the Euclid-Euler perfect-number pairing when applicable.

First Twelve Known Mersenne Primes

#Exponent \(p\)\(M_p = 2^p - 1\)DigitsDiscovered
1231Ancient
2371Ancient
35312Ancient
471273Ancient
5138,19141456 (anon.)
617131,07161588 Cataldi
719524,28761588 Cataldi
8312,147,483,647101772 Euler
9612.3 × 10^18191883 Pervushin
10896.2 × 10^26271911 Powers
111071.6 × 10^32331914 Powers
121271.7 × 10^38391876 Lucas

The GIMPS Project

The Great Internet Mersenne Prime Search (GIMPS), launched in 1996 by George Woltman, is a distributed-computing project where volunteers donate CPU time to run Lucas-Lehmer tests on candidate exponents. As of 2024, every Mersenne prime since M_35 = M_{1398269} (1996) has been discovered by GIMPS. A single Lucas-Lehmer test at the modern frontier (exponents near \(10^8\)) takes weeks of GPU computation.

Fun Facts About Mersenne Primes

Frequently Asked Questions

What is a Mersenne prime?

A Mersenne prime is a prime number of the form \(2^p - 1\), where \(p\) is also prime. The first few are 3, 7, 31, 127, and 8,191. As of 2024, 52 Mersenne primes are known; the largest known prime (\(M_{136{,}279{,}841}\)) is a Mersenne prime with over 41 million digits.

How does the Lucas-Lehmer test work?

For a prime exponent \(p \geq 3\), define \(S_0 = 4\) and \(S_i = S_{i-1}^2 - 2 \pmod{M_p}\). The Mersenne number \(M_p = 2^p - 1\) is prime if and only if \(S_{p-2} \equiv 0 \pmod{M_p}\). The test runs in \(p - 2\) iterations, each a single modular squaring.

Why must \(p\) be prime?

If \(p = ab\) with both factors greater than 1, then \(2^p - 1\) is divisible by \(2^a - 1\) (and by \(2^b - 1\)), so \(M_p\) is composite. The converse is not true: \(p\) being prime does not imply \(M_p\) is prime. For example \(p = 11\) is prime but \(M_{11} = 2047 = 23 \times 89\) is composite.

What is the connection between Mersenne primes and perfect numbers?

The Euclid-Euler theorem states that every even perfect number has the form \(2^{p-1}(2^p - 1)\) where \(2^p - 1\) is a Mersenne prime. So every Mersenne prime generates exactly one even perfect number, and every even perfect number comes from a Mersenne prime. Whether any odd perfect numbers exist is one of the oldest open problems in mathematics.

Why does \(M_p\) have \(p\) consecutive 1-bits in binary?

The number \(2^p\) in binary is a 1 followed by \(p\) zeros. Subtracting 1 converts all \(p\) trailing zeros into 1s. So \(2^p - 1\) in binary is exactly \(p\) ones — the defining visual signature of every Mersenne number, prime or composite.

What is the largest exponent this tool can test?

This tool tests exponents up to 5,000 so the Lucas-Lehmer iteration completes within a normal web request. For larger exponents (including the GIMPS frontier near \(10^8\)), dedicated software such as Prime95 is required since a single test can take weeks of compute time on a modern GPU.

Additional Resources

Reference this content, page, or tool as:

"Mersenne Prime Checker" at https://MiniWebtool.com/mersenne-prime-checker/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Apr 18, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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