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A prime p satisfying the congruence 2 p−1 ≡ 1 (mod p 2) is called a Wieferich prime. Although the number of Wieferich primes is believed to be infinite, the only ones that have been discovered so far are 1093 and 3511. This paper describes a search for further solutions. The search was conducted via a large scale Internet based computation. The result that… (More)

To supplement existing data, solutions of a p−1 ≡ 1 (mod p 2) are tabulated for primes a, p with 100 < a < 1000 and 10 4 < p < 10 11. For a < 100, five new solutions p > 2 32 are presented. One of these, p = 188748146801 for a = 5, also satisfies the " reverse " congruence p a−1 ≡ 1 (mod a 2). An effective procedure for searching for such " double solutions… (More)

Using a carefully optimized segmented sieve and an efficient checking algorithm, the Goldbach conjecture has been verified and is now known to be true up to 4 · 10 14. The program was distributed to various workstations. It kept track of maximal values of the smaller prime p in the minimal partition of the even numbers, where a minimal partition is a… (More)

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