Displaying similar documents to “Congruences involving F-partition functions.”

Congruences involving the Fermat quotient

Romeo Meštrović (2013)

Czechoslovak Mathematical Journal

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Let p > 3 be a prime, and let q p ( 2 ) = ( 2 p - 1 - 1 ) / p be the Fermat quotient of p to base 2 . In this note we prove that ∑ k = 1 p - 1 1 k · 2 k ≡ q p ( 2 ) - p q p ( 2 ) 2 2 + p 2 q p ( 2 ) 3 3 - 7 48 p 2 B p - 3 ( mod p 3 ) , which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that q p ( 2 ) 3 ≡ - 3 ∑ k = 1 p - 1 2 k k 3 + 7 16 ∑ k = 1 ( p - 1 ) / 2 1 k 3 ( mod p ) , which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum ∑ k = 1 p - 1 1 / ( k 2 · 2 k ) modulo p 2 that also generalizes...

Kummer congruences for expressions involving generalized Bernoulli polynomials

Glenn J. Fox (2002)

Journal de théorie des nombres de Bordeaux

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We illustrate how a particular expression, involving the generalized Bernoulli polynomials, satisfies systems of congruence relations if and only if a similar expression, involving the generalized Bernoulli numbers, satisfies the same congruence relations. These congruence relations include the Kummer congruences, and recent extensions of the Kummer congruences provided by Gunaratne.