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Algorithmes pour vérifier la conjecture de Syracuse

Jacques Arsac (1987)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Algorithmus für Bell'sche Polynome

Robert Karpe (1972)

Archivum Mathematicum

All Congruent Numbers Less than 2000.

Gerhard Kramarz (1985/1986)

Mathematische Annalen

Almost powers in the Lucas sequence

Yann Bugeaud, Florian Luca, Maurice Mignotte, Samir Siksek (2008)

Journal de Théorie des Nombres de Bordeaux

The famous problem of determining all perfect powers in the Fibonacci sequence ${(F_{n})}_{n \geq 0}$ and in the Lucas sequence ${(L_{n})}_{n \geq 0}$ has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations $L_{n} = q^{a} y^{p}$ , with $a > 0$ and $p \geq 2$ , for all primes $q < 1087$ and indeed for all but $13$ primes $q < 10^{6}$ . Here the strategy of [10] is not sufficient due to the sizes of...

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 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...

An identity generator: basic commutators.

Farrokhi D.G., M. (2008)

The Electronic Journal of Combinatorics [electronic only]

An inequality for Fibonacci numbers

Horst Alzer, Florian Luca (2022)

Mathematica Bohemica

We extend an inequality for Fibonacci numbers published by P. G. Popescu and J. L. Díaz-Barrero in 2006.

An introduction to finite fibonomial calculus

Ewa Krot (2004)

Open Mathematics

This is an indicatory presentation of main definitions and theorems of Fibonomial Calculus which is a special case of ψ-extented Rota's finite operator calculus [7].