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## Colloquium Mathematicum

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## Even partition functions and $2$-adic analysis

### Volume 173 / 2023

Colloquium Mathematicum 173 (2023), 175-195 MSC: Primary 11P83; Secondary 11D88, 11L05, 11L40. DOI: 10.4064/cm9000-2-2023 Published online: 8 May 2023

#### Abstract

Let $\mathcal A$ denote a set of positive integers, and let $p(\mathcal A,n)$ denote the associated partition function. Let $\beta$ be an odd positive integer, and let $P(z)$ be a polynomial in $\mathbb F_2[z]$ of order $\beta$ such that $P(0)=1$. J.-L. Nicolas, I. Z. Ruzsa and A. Sárközy proved that there exists a unique set $\mathcal A=\mathcal A(P)$ such that $\sum _{n\geq 0}p(\mathcal A,n)z^n\equiv P(z) \pmod 2$; that is, the partition function $p(\mathcal {A},n)$ is even from a certain point on. The problem of determining the elements of the set $\mathcal A(P)$ is not an easy one and several particular cases have already been studied; namely, when $P$ is irreducible and $\beta$ is a prime number $p$ such that the order of $2$ modulo $p$ is $p-1$, $(p-1)/2$, $(p-1)/3$ or $(p-1)/4$. In this paper, we consider the case where $P$ is irreducible such that the order of $2$ modulo $\beta$ is $\varphi (\beta )/2$ where $\varphi$ is Euler’s function.

#### Authors

• Naceur BaccarISITCOM Hammam Sousse
Université de Sousse
4011 Hammam Sousse, Tunisie
e-mail
• Jean-Louis NicolasUniv Lyon
Université Claude Bernard, Lyon 1
CNRS UMR 5208
Institut Camille Jordan, Mathématiques
F-69622 Villeurbanne, France
e-mail

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