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Displaying 41 – 60 of 242

An elementary proof of a congruence by Skula and Granville

Romeo Meštrović (2012)

Archivum Mathematicum

Let $p \geq 5$ be a prime, and let $q_{p} (2) : = (2^{p - 1} - 1) / p$ be the Fermat quotient of $p$ to base $2$ . The following curious congruence was conjectured by L. Skula and proved by A. Granville $q_{p} {(2)}^{2} \equiv - \sum_{k = 1}^{p - 1} \frac{2^{k}}{k^{2}} (mod p) .$ In this note we establish the above congruence by entirely elementary number theory arguments.

An explicit evaluation of the Gosper sum.

Cimadevilla Villacorta, Jorge Luis (2010)

Integers

An extension of the Lucas theorem

Jacques Boulanger, J.-L. Chabert (2001)

Acta Arithmetica

An infinite family of dual sequence identities.

Chapman, Robin (2005)

Integers

An observation on the extension of Abel's lemma.

Patkowski, Alexander E. (2010)

Integers

Another simple proof of the quintuple product identity.

Chan, Hei-Chi (2005)

International Journal of Mathematics and Mathematical Sciences

Asymptotic behavior of Pascal's triangle modulo a prime

Brad Wilson (1998)

Acta Arithmetica

Asymptotic expansion of the equipoise curve of a polynomial inequality.

Eggleton, Roger B., Galvin, William P. (2002)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Asymptotic expansions of certain sums involving powers of binomial coefficients.

Zhao, Chun-Xue, Zhao, Feng-Zhen (2010)

Journal of Integer Sequences [electronic only]

Asymptotic order of the square-free part of $N!$ .

Broughan, Kevin A. (2002)

Integers

Bijective proofs of parity theorems for partition statistics.

Shattuck, Mark (2005)

Journal of Integer Sequences [electronic only]

Binomial coefficient predictors.

Shevelev, Vladimir (2011)

Journal of Integer Sequences [electronic only]

Bounds for the counting function of the Jordan-Pólya numbers

Jean-Marie De Koninck, Nicolas Doyon, A. Arthur Bonkli Razafindrasoanaivolala, William Verreault (2020)

Archivum Mathematicum

A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$ .

In this paper, we apply character sum estimates to study products of factorials modulo $p$ .

Currently displaying 41 – 60 of 242

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Displaying 41 – 60 of 242

An elementary proof of a congruence by Skula and Granville

An explicit evaluation of the Gosper sum.

An extension of the Lucas theorem

An infinite family of dual sequence identities.

An observation on the extension of Abel's lemma.

Another simple proof of the quintuple product identity.

Asymptotic behavior of Pascal's triangle modulo a prime

Asymptotic expansion of the equipoise curve of a polynomial inequality.

Asymptotic expansions of certain sums involving powers of binomial coefficients.

Asymptotic order of the square-free part of $N!$ .

Bijective proofs of parity theorems for partition statistics.

Binomial coefficient predictors.

Bounds for the counting function of the Jordan-Pólya numbers

Brownian motion and the generalized Catalan numbers.

Catalan numbers, the Hankel transform, and Fibonacci numbers.

Certain sums involving inverses of binomial coefficients and some integrals.

Character sums and products of factorials modulo $p$

Combinatorial congruences and Stirling numbers

Combinatorial interpretations of a generalization of the Genocchi numbers.

Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences.

Currently displaying 41 – 60 of 242