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$p$ -adic analysis and Bell numbers of two variables. (Analyse $p$ -adique et nombres de Bell à deux variables.)

Mazouz, Abdelhak (1996)

Bulletin of the Belgian Mathematical Society - Simon Stevin

p-adic Dedekind sums.

Kenneth H. Rosen, William M. Snyder (1985)

Journal für die reine und angewandte Mathematik

p-adic valuations of some sums of multinomial coefficients

Zhi-Wei Sun (2011)

Acta Arithmetica

Padovan and Perrin numbers as products of two generalized Lucas numbers

Kouèssi Norbert Adédji, Japhet Odjoumani, Alain Togbé (2023)

Archivum Mathematicum

Let $P_{m}$ and $E_{m}$ be the $m$ -th Padovan and Perrin numbers respectively. Let $r, s$ be non-zero integers with $r \geq 1$ and $s \in {- 1, 1}$ , let ${U_{n}}_{n \geq 0}$ be the generalized Lucas sequence given by $U_{n + 2} = r U_{n + 1} + s U_{n}$ , with $U_{0} = 0$ and $U_{1} = 1 .$ In this paper, we give effective bounds for the solutions of the following Diophantine equations $P_{m} = U_{n} U_{k} and E_{m} = U_{n} U_{k},$ where $m$ , $n$ and $k$ are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.

Padua and Pisa are exponentially far apart.

Benjamin M. M. De Weger (1997)

Publicacions Matemàtiques

We answer the question posed by Ian Stewart which Padovan numbers are at the same time Fibonacci numbers. We give a result on the difference between Padovan and Fibonacci numbers, and on the growth of Padovan numbers with negative indices.

Parallelepipeds, nilpotent groups and Gowers norms

Bernard Host, Bryna Kra (2008)

Bulletin de la Société Mathématique de France

In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions $2$ and $3$ and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.