Sets with even partition functions and 2-adic integers. II.

Baccar, N. ; Zekraoui, A.

Journal of integer sequences, Tome 13 (2010) no. 1

Résumé

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 $ _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$.

Détail

Classification : 11P83, 11B50, 11D88, 11G20
Keywords: partitions, periodic sequences, order of a polynomial, cyclotomic polynomials, resultant, 2-adic integers, elliptic curves