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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 introduce an infinite class of polynomial sequences a t (n; z) with integer parameter t 1, which reduce to the well-known Stern (diatomic) sequence when z = 1 and are (0, 1)-polynomials when t 2. Using these polynomial sequences, we derive two different characterizations of all hyperbinary expansions of an integer n 1. Furthermore, we study the… (More)

1. INTRODUCTION. Two of the most ubiquitous objects in mathematics are the sequence of prime numbers and the binomial coefficients (and thus Pascal's triangle). A connection between the two is given by a well-known characterization of the prime numbers:

An analogue for composite moduli m ≥ 2 of the Wilson quotient is studied. Various congruences are derived, and the question of when these quotients are divisible by m is investigated; such an m will be called a " Wilson number ". It is shown that numbers in certain infinite classes cannot be Wilson numbers. Eight new Wilson numbers up to 500 million were… (More)

We study congruence and divisibility properties of a class of com-binatorial sums that involve products of powers of two binomial coefficients, and show that there is a close relationship between these sums and the theorem of Wolstenholme. We also establish congruences involving Bernoulli numbers, and finally we prove that under certain conditions the sums… (More)