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 $p$.