Introduction to finite fields and their applications
Introduction to finite fields and their applications
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Let N be the set of positive integers and A a subset of N. For n ∈ N, let p(A, n) denote the number of partitions of n with parts in A. In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any N ∈ N and B ⊂ {1, 2,...,N}, there is a unique set A = A0(B, N), such that p(A, n) is even for n N. Soon after, Ben Saïd and Nicolas (Acta Arith. 106 (2003) 183) considered σ (A, n) = Σd\n,d ∈ Ad, and proved that for all k ≥ 0, the sequence (σ(A, 2kn) mod 2k+1)n≥1 is periodic on n. In this paper, we generalise the above works for any formal power series f in F2[z] with f (0) = 1, by constructing a set A such that the generating function fA of A is congruent to f modulo 2, and by showing that if f = P/Q, where P and Q are in F2[z] with P(0) = Q(0) = 1, then for all k ≥ 0 the sequence (σ (A, 2kn) mod 2k+1)n≥1 is periodic on n.