Let l greater than or equal to 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 recursionx(0) = 4, x(k) = x(k-1)(2) - 2Then p is a prime if and only if each x(k) is relatively prime to p, for 0 less than or equal to k less than or equal to 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(root3). 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 recursionx(0) = -2 x(k) = (x(k-1)(2) + 12)(2) a/ 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 less than or equal to k less than or equal to 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 byy(2) = x(3) - 12x.We provide the in section 2.
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