2 7 M ar 2 00 3 Bernoulli Numbers , Wolstenholme ’ s Theorem , and p 5 Variations of Lucas ’ Theorem ∗
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چکیده
In [7, Proposition 2.9] we further proved that H1(p − 1) ≡ 0 (mod p ) if and only if p divides the numerator of Bp(p−1)−2, which never happens for primes less than 12 million. There is another important equivalent statement of Wolstenholme’s Theorem by using combinatorics. D.F. Bailey [1] generalizes it to the following form. Theorem 1.1. ([1, Theorem 4]) Let n and r be non-negative integers and p ≥ 5 be a prime. Then
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6 M ar 2 00 3 Bernoulli Numbers , Wolstenholme ’ s Theorem , and p 5 Variations of Lucas ’ Theorem ∗
In [7] we further proved that H1(p−1) ≡ 0 (mod p ) if and only if p divides the numerator of Bp−3, which never happens for primes less than 12 million [2]. There is another important equivalent statement of Wolstenholme’s Theorem by using combinatorics. D.F. Bailey [1] generalizes it to the following form. Theorem 1.1. ([1, Theorem 4]) Let n and r be non-negative integers and p ≥ 5 be a prime. ...
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In this note we shall improve some congruences of G.S. Kazandzidis and D.F. Bailey to higher prime power moduli, by studying the relation between irregular pairs of the form (p,p− 3) and a refined version of Wolstenholme’s theorem. © 2006 Elsevier Inc. All rights reserved. MSC: primary 11A07, 11Y40; secondary 11A41, 11M41
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