Abstract

For P ∈ 𝔽_{2}[z] with P (0) = 1 and deg(P) ≥ 1, let A = A(P) be the unique subset of ℕ such that ∑_{n≥0} p(A, n)z^{n} ≡ P (z) (mod 2), where p(A, n) is the number of partitions of n with parts in 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 𝔽_{2}[z]. N. Baccar proved that the elements of A(P) of the form 2^{k}m, where k ≥ 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}} ≡ 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.