Journal of Integer Sequences, Vol. 13 (2010), Article 10.1.3

Sets with Even Partition Functions and 2-adic Integers II


N. Baccar
Université de Sousse
ISITCOM Hammam Sousse
Dép. de Math Inf.
5 Bis Rue 1 Juin 1955
4011 Hammam Sousse
Tunisie

A. Zekraoui
Université de Monastir
F. S. M.
Dép. de Math.
Avenue de l'environnement
5000 Monastir
Tunisie

Abstract:

For $ P\in \mathbb{F}_2[z]$ with $ P(0)=1$ and $ \deg(P)\geq 1$, let $ {\cal A}={\cal A}(P)$ be the unique subset of $ \mathbb{N}$ such that $ \sum_{n\geq 0}p({\cal A},n)z^n\equiv P(z)$ (mod $ 2$), where $ p({\cal A},n)$ is the number of partitions of $ n$ with parts in $ {\cal A}$. Let $ p$ be an odd prime number, and let $ P$ be irreducible of order $ p$ ; i.e., $ p$ is the smallest positive integer such that $ P$ divides $ 1+z^p$ in $ \mbox{$\mathbb{F}$}$$ _2[z]$. N. Baccar proved that the elements of $ {\cal A}(P)$ of the form $ 2^km$, where $ k\geq 0$ and $ m$ is odd, are given by the $ 2$-adic expansion of a zero of some polynomial $ R_m$ with integer coefficients. Let $ s_p$ be the order of $ 2$ modulo $ p$, i.e., the smallest positive integer such that $ 2^{s_p}\equiv 1$ (mod $ p$). Improving on the method with which $ R_m$ was obtained explicitly only when $ s_p=\frac{p-1}{2}$, here we make explicit $ R_m$ when $ s_p=\frac{p-1}{3}$. For that, we have used the number of points of the elliptic curve $ x^3+ay^3 =1 $ modulo $ p$.

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Received July 16 2009; revised version received December 23 2009. Published in Journal of Integer Sequences, December 31 2009.

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