An elliptic curve test for Mersenne primes

Let l ≥ 3 be a prime, and let p = 2 − 1 be the corresponding Mersenne number. The Lucas-Lehmer test for the primality of p goes as follows. Define the sequence of integers xk by the recursion x0 = 4, xk = x 2 k−1 − 2. Then p is a prime if and only if each xk is relatively prime to p, for 0 ≤ k ≤ l − 3, and gcd(xl−2, p) > 1. We show, in the first section, 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 xk by the recursion x0 = −2, xk = (x2k−1 + 12) 2