EuDML | Browse

Items

All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Displaying 261 – 280 of 352

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 estimate for the number of reducible Bessel polynomials of bounded degree

M. Filaseta, S. Graham (1993)

Colloquium Mathematicae

An "exact" formula for the 2n-th Bernoulli number

Hans Riesel (1975)

Acta Arithmetica

An "exact" formula for the m-th Bernoulli number

S. Chowla, P. Hartung (1972)

Acta Arithmetica

An explicit evaluation of the Gosper sum.

Cimadevilla Villacorta, Jorge Luis (2010)

Integers

An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers

Carlos Alexis Gómez Ruiz, Florian Luca (2014)

Colloquium Mathematicae

A generalization of the well-known Fibonacci sequence ${F ₙ}_{n \geq 0}$ given by F₀ = 0, F₁ = 1 and $F_{n + 2} = F_{n + 1} + F ₙ$ for all n ≥ 0 is the k-generalized Fibonacci sequence ${F ₙ^{(k)}}_{n \geq - (k - 2)}$ whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula $F ₙ ² + F ²_{n + 1} ² = F_{2 n + 1}$ holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This generalizes...