Class Numbers and Self-Conjugate 7-Cores
Ken Ono, Wissam Raji

TL;DR
This paper explores the enumeration of self-conjugate 7-core partitions, revealing their connection to Hurwitz class numbers and deriving formulas involving prime powers and discriminants.
Contribution
It establishes a novel relationship between self-conjugate 7-core counts and Hurwitz class numbers, extending understanding of partition theory and modular forms.
Findings
sc_7(n)=0 for n ≡ 7 mod 8
sc_7(n) relates to Hurwitz class numbers for certain n
Derived formulas for sc_7(n) involving prime powers and discriminants
Abstract
We investigate , the number of self-conjugate -core partitions of size . It turns out that for . For , with we find that is essentially a Hurwitz class number. Using recent work of Gao and Qin, we show that where and . This fact implies several corollaries which are of interest. For example, if is a fundamental discriminant and is a prime with , then for every positive integer we have where is the Legendre symbol.
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Advanced Combinatorial Mathematics
