Mersenne Prime Checker

Test whether 2^p - 1 is a Mersenne prime for a given exponent p. Uses the Lucas-Lehmer test with an animated iteration trace, a binary bit-pattern visual, Euclid-Euler perfect-number pairing and the 52 known Mersenne primes.

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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:

$M_2 = 3$
$M_3 = 7$
$M_5 = 31$
$M_7 = 127$
$M_{13} = 8{,}191$
$M_{17} = 131{,}071$
$M_{19} = 524{,}287$
$M_{31} = 2{,}147{,}483{,}647$ (found by Euler in 1772 — the largest known prime for 104 years)

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

Enter an exponent $p$: any positive integer from 1 to 5,000.
Click Check: the tool first checks whether $p$ is prime; if not, it explains why $M_p$ must be composite.
For prime $p$: the Lucas-Lehmer recurrence runs $p - 2$ iterations modulo $M_p$.
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 $p$	$M_p = 2^p - 1$	Digits	Discovered
1	2	3	1	Ancient
2	3	7	1	Ancient
3	5	31	2	Ancient
4	7	127	3	Ancient
5	13	8,191	4	1456 (anon.)
6	17	131,071	6	1588 Cataldi
7	19	524,287	6	1588 Cataldi
8	31	2,147,483,647	10	1772 Euler
9	61	2.3 × 10^18	19	1883 Pervushin
10	89	6.2 × 10^26	27	1911 Powers
11	107	1.6 × 10^32	33	1914 Powers
12	127	1.7 × 10^38	39	1876 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

$M_{31} = 2{,}147{,}483{,}647$ is the largest 32-bit signed integer — the famous $\texttt{INT\_MAX}$ in C. This is no coincidence: the value comes from the fact that $M_{31}$ is prime and therefore a natural "just-short-of-overflow" boundary.
There are gaps of unknown size between successive Mersenne primes. It is not known whether there are infinitely many Mersenne primes — the Lenstra-Pomerance-Wagstaff conjecture predicts there are, growing roughly like $e^\gamma \log_2 p$.
In 2008, the Electronic Frontier Foundation awarded US$100,000 to the first discoverer of a 10-million-digit prime. The prize went to UCLA's GIMPS team for $M_{43112609}$. A US$150,000 prize is still available for the first 100-million-digit prime.
$M_{31}$ appears on the 1811 Russian 100-ruble commemorative banknote honoring Euler's discovery — one of the few prime numbers ever printed on currency.
Because every Mersenne prime yields a perfect number, humanity has exactly 52 even perfect numbers on file (matching the 52 known 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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