Mersenne prime: M(p) = 2ᵖ−1 where p is prime and M(p) is prime. Known: M(2)=3, M(3)=7, M(5)=31, M(7)=127... 51 known as of 2024. Each Mersenne prime gives a perfect number: 2ᵖ⁻¹(2ᵖ−1). Lucas-Lehmer test: efficient primality proof.
How to Use
Enter n:
- M(n): 2ⁿ−1
- Prime?: Lucas-Lehmer test
- Perfect: Associated number
Lucas-Lehmer Test
For odd prime p: define s₀=4, sₙ=sₙ₋₁²−2 (mod M(p)). M(p) is prime iff s_{p-2}≡0 (mod M(p)). Runs in O(p² log p) time. The most efficient known primality test for Mersenne numbers.
GIMPS Project
Great Internet Mersenne Prime Search: distributed computing since 1996. Has found 17 of the 51 known Mersenne primes. The largest known prime (2024): 2^136,279,841−1 (41,024,320 digits). Prizes available for discoveries.
Step-by-Step Instructions
- 1Enter p.
- 2Check if 2ᵖ−1 prime.
- 3View Lucas-Lehmer.
- 4Get perfect number.
- 5See known primes.