A Mersenne prime is a prime number that can be written in the form M_p = 2^p − 1, where p is itself a prime number. They are named after the French monk Marin Mersenne who studied them in the 17th century. Not all values of prime p yield a Mersenne prime — for example, p=11 gives 2047 = 23 × 89, which is composite.

The Lucas–Lehmer primality test is a deterministic algorithm specifically for Mersenne numbers. For an odd prime p, define the sequence: s₀ = 4, and s_i = s_i−1² − 2 (mod M_p). M_p is prime if and only if s_p−2 ≡ 0 (mod M_p). This test is remarkably efficient — it runs in O(p² log p) time — making it the standard method for verifying Mersenne primes.

No. While p being prime is a necessary condition for M_p to be prime, it is not sufficient. If p is composite, then 2^p−1 is definitely composite. However, many prime p values produce composite Mersenne numbers. The first counterexample is p=11: 2¹¹−1 = 2047 = 23 × 89. Other examples include p=23, p=29, and p=37.

As of 2024, there are 51 known Mersenne primes. The largest is M_82589933, discovered in December 2018 by the GIMPS (Great Internet Mersenne Prime Search) project. It has 24,862,048 decimal digits and is also the largest known prime number overall. It is widely believed that there are infinitely many Mersenne primes, though this remains unproven.

GIMPS (Great Internet Mersenne Prime Search) is a collaborative distributed computing project where volunteers worldwide contribute their computers' processing power to search for new Mersenne primes. Founded in 1996, GIMPS has discovered the last 17 Mersenne primes. Anyone can participate by downloading the free Prime95 software from mersenne.org.

Mersenne primes have several important applications: they are used in pseudorandom number generators (like the Mersenne Twister), in cryptography for testing primality of large numbers, and in error-correcting codes. Historically, the search for Mersenne primes has also driven advances in distributed computing, algorithm design, and computer hardware testing. The largest known prime number has almost always been a Mersenne prime for the past century.

Yes, for small exponents. This online tool uses the Lucas–Lehmer test implemented in JavaScript with BigInt arithmetic, which works well for p up to about 5,000–10,000. For larger exponents (like those used in GIMPS), specialized software like Prime95 using Fast Fourier Transforms and optimized assembly is required — a single test for p ≈ 100,000,000 can take weeks on a consumer PC. This tool is best suited for educational purposes and verifying smaller Mersenne primes.

The Mersenne Twister is a widely-used pseudorandom number generator (PRNG) named after its period length, which is a Mersenne prime: 2¹⁹⁹³⁷ − 1. Developed in 1997, it is the default PRNG in many programming languages including Python, Ruby, and R. Despite its name, the algorithm itself is not directly related to Mersenne prime discovery — it simply uses a Mersenne prime as its period length for excellent statistical properties.