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An odd prime p is called a Wieferich prime if 2 p−1 ≡ 1 (mod p 2); alternatively, a Wilson prime if (p − 1)! ≡ −1 (mod p 2). To date, the only known Wieferich primes are p = 1093 and 3511, while the only known Wilson primes are p = 5, 13, and 563. We report that there exist no new Wieferich primes p < 4 × 10 12 , and no new Wilson primes p < 5 × 10 8. It is(More)
The generalized harmonic numbers H (k) n = n j=1 j −k satisfy the well-known congruence H (k) p−1 ≡ 0 (mod p) for all primes p ≥ 3 and integers k ≥ 1. We derive q-analogs of this congruence for two different q-analogs of the sum H (k) n. The results can be written in terms of certain determinants of binomial coefficients which have interesting properties in(More)
We report the discovery of a new factor for each of the Fermat numbers F 13 , F 15 , F 16. These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with 2391, 9808 and 19694(More)