Displaying similar documents to “Mod p³ analogues of theorems of Gauss and Jacobi on binomial coefficients”

Strong versions of Kummer-type congruences for Genocchi numbers and polynomials and tangent coefficients

Mehmet Cenkci (2005)

Acta Mathematica Universitatis Ostraviensis

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We use the properties of p -adic integrals and measures to obtain general congruences for Genocchi numbers and polynomials and tangent coefficients. These congruences are analogues of the usual Kummer congruences for Bernoulli numbers, generalize known congruences for Genocchi numbers, and provide new congruences systems for Genocchi polynomials and tangent coefficients.

Some congruences for 3-component multipartitions

Tao Yan Zhao, Lily J. Jin, C. Gu (2016)

Open Mathematics

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Let p3(n) denote the number of 3-component multipartitions of n. Recently, using a 3-dissection formula for the generating function of p3(n), Baruah and Ojah proved that for n ≥ 0, p3(9n + 5) ≡ 0 (mod 33) and p3 (9n + 8) ≡ 0 (mod 34). In this paper, we prove several congruences modulo powers of 3 for p3(n) by using some theta function identities. For example, we prove that for n ≥ 0, p3 (243n + 233) ≡ p3 (729n + 638) ≡ 0 (mod 310).

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.

The terms of the form 7kx² in the generalized Lucas sequence with parameters P and Q

Olcay Karaatlı (2016)

Acta Arithmetica

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Let Vₙ(P,Q) denote the generalized Lucas sequence with parameters P and Q. For all odd relatively prime values of P and Q such that P² + 4Q > 0, we determine all indices n such that Vₙ(P,Q) = 7kx² when k|P. As an application, we determine all indices n such that the equation Vₙ = 21x² has solutions.