Abstract :
Let ℓ 3 be a prime, and let p=2ℓ-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 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 Section 1, that this test is based on the successive squaring of a point on the one-dimensional algebraic torus T over , associated to the real quadratic field . 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 then we show that p is prime if and only if is relatively prime to p, for 0 k ℓ-2, and gcd(xℓ-1,p)>1. This test involves the successive squaring of a point on the elliptic curve E over defined by y2=x3-12x.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?
