Deterministic elliptic curve primality proving for a special sequence of numbers

We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit complexity that is quasi-quadratic in log N. Notably, neither of the classical "N-1" or "N+1" primality tests apply to the integers in our sequence. We discuss how this algorithm may be applied, in combination with sieving techniques, to efficiently search for very large primes. This has allowed us to prove the primality of several integers with more than 100,000 decimal digits, the largest of which has more than a million bits in its binary representation. At the time it was found, it was the largest proven prime N for which no significant partial factorization of N-1 or N+1 is known.Comment: 16 pages, corrected a minor sign error in 5.

Mersenne primes and class field theory

The Lucas-Lehmer-test is used to find Mersenne primes. In the thesis a question of Lehmer about this test is answered.UBL - phd migration 201

Could, or should, the ancient Greeks have discovered the Lucas-Lehmer test?

The Lucas-Lehmer (LL) test is the most efficient known for testing the primality of Mersenne numbers, i.e. the integers Ml = 2l − 1, for l ≥ 1. The Mersenne numbers are so-called in honour of the French scholar Marin Mersenne (1588-1648), who in 1644 published a list of exponents l ≤ 257 which he conjectured produced all and only those Ml which are prime, for l in this range, namely l = 2,3,5,7, 13, 17, 19,31,67, 127 and 257. Mersenne's list turned out to be incorrect, omitting the prime-producing l = 61, 89 and 107 and including the composite-producing l = 67 and 257, although this was not finally confirmed until 1947, using both the LL test and contemporary mechanical calculators