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Sets with even partition functions and cyclotomic numbers. (English) Zbl 1443.11217

Summary: Let \(P\in \mathbb{F}_2[z]\) be such that \(P(0)=1\) and \(\text{degree}(P)\geq 1\). J. L. Nicolas et al. [J. Number Theory 73, No. 2, 292–317 (1998; Zbl 0921.11050)] proved that there exists a unique subset \(\mathcal{A}=\mathcal{A}(P)\) of \(\mathbb{N}\) such that \(\sum_{n\geq 0}p(\mathcal{A},n)z^{n}\equiv P(z)\bmod 2\), where \(p(\mathcal A,n)\) is the number of partitions of \(n\) with parts in \(\mathcal A\). Let \(m\) be an odd positive integer and let \(\chi(\mathcal A)\) be the characteristic function of the set \(\mathcal A\). Finding the elements of the set \(\mathcal{A}\) of the form \(2^k m\), \(k\geq 0\), is closely related to the \(2\)-adic integer \[ S(\mathcal A,m)=\chi(\mathcal A,m)+2\chi(\mathcal A,2m)+4\chi(\mathcal A,4m)+\cdots =\sum_{k=0}^{\infty}2^k \chi(\mathcal A,2^k m), \] which has been shown to be an algebraic number. Let \(G_{m}\) be the minimal polynomial of \(S(\mathcal A,m)\). In precedent works there were treated the case \(P\) irreducible of odd prime order \(p\). In this setting, taking \(p=1+ef\), where \(f\) is the order of \(2\) modulo \(p\), explicit determinations of the coefficients of \(G_m\) have been made for \(e=2\) and 3. In this paper, we treat the case \(e=4\) and use the cyclotomic numbers to make explicit \(G_m\).

MSC:

11P83 Partitions; congruences and congruential restrictions
11B50 Sequences (mod \(m\))
11C08 Polynomials in number theory

Citations:

Zbl 0921.11050
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References:

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