The Lucas-Lehmer Test For Mersenne Primes

A Mersenne prime is in the form of 2^S−1. Known Mersenne prime numbers are 2³−1, 2⁵−1, 2⁷−1, 2¹³−1, 2¹⁷−1, 2¹⁹−1, 2³¹−1, 2⁶¹−1, 2⁸⁹−1, 2¹⁰⁷−1 and 2¹²⁷−1. Overall Mersenne primes are efficient in their implementation. This article outlines the the Lucas-Lehmer test to test for a Mersenne prime number. A fast elliptic curve name FourQ uses a prime of 2¹²⁷-1:

FourQ: A Story of Elliptic Curves Microsoft Research and Cloudflare

Lucas-Lehmer Test

The Lucas–Lehmer test (LLT) is used to test for the primality of Mersenne numbers. It was initally created Edouard Lucas and then enhanced by Derrick Hentry Lehmer. For this test, we initially define u_0=0 and we calculate:

u_{k+1}=(u_k²−2)(mod n)

If u_{s−2}=0 then it is a prime number. Here are some Mersenne Primes:

• 2³−1 Try.
• 2⁵−1 Try.
• 2⁷−1 Try.
• 2¹³−1 Try.
• 2¹⁷−1 Try.
• 2¹⁹−1 Try.
• 2³¹−1 Try.
• 2⁶¹−1 Try.
• 2⁸⁹−1 Try.