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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)
This test paper is typeset in the document class amsart, without any additional style files. Font size and text size are easy to change; here they are 11pt and 5.5 " × 8.5 " , respectively. This appears to be quite close to the present format of the Quarterly—?. If we decide to have abstracts for articles in the Fibonacci Quarterly, which I think is a good… (More)
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 show that the resultants with respect to x of certain linear forms in Chebyshev polynomials with argument x are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of… (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:
We report the discovery of new 27-decimal digit factors of the thirteenth and sixteenth Fermat numbers. Each of the new factors was found by the elliptic curve method. After division by the new factors and other known factors, the quotients are seen to be composite numbers with 2391 and 19694 decimal digits respectively.