An elliptic curve test for Mersenne primes

Abstract

Let ℓ ⩾ 3 be a prime, and let p = 2 ℓ - 1 be the corresponding Mersenne number. The Lucas–Lehmer test for the primality of p goes as follows. Define the sequence of integers x k by the recursion x 0 = 4 , x k = x k - 1 2 - 2 . Then p is a prime if and only if each x k is relatively prime to p, for 0 ⩽ k ⩽ ℓ - 3 , and gcd ( x ℓ - 2 , p ) > 1 . We show, in the Section 1, that this test is based on the successive squaring of a point on the one-dimensional algebraic torus T over Q , associated to the real quadratic field k = Q ( 3 ) . This suggests that other tests could be developed, using different algebraic groups. As an illustration, we will give a second test involving the sucessive squaring of a point on an elliptic curve. If we define the sequence of rational numbers x k by the recursion x 0 = - 2 , x k = ( x k - 1 2 + 12 ) 2 4 · x k - 1 · ( x k - 1 2 - 12 ) , then we show that p is prime if and only if x k · ( x k 2 - 12 ) is relatively prime to p, for 0 ⩽ k ⩽ ℓ - 2 , and gcd ( x ℓ - 1 , p ) > 1 . This test involves the successive squaring of a point on the elliptic curve E over Q defined by y 2 = x 3 - 12 x . We provide the details in Section 2. The two tests are remarkably similar. For example, both take place on groups with good reduction away from 2 and 3. Can one be derived from the other?