New Congruences of Partitions With Odd Parts Distinct

2014/07/21 by Liuquan Wang, Wang, Liuquan
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 05A17 #Secondary 11P83

paper · pdf · doi:10.48550/arxiv.1407.5436

openalex publication_date 2014/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let pod(n) denote the number of partitions of n with odd parts distinct, and r_k(n) be the number of representations of n as sum of k squares. We find the following two arithmetic relations: for any integer n≥ 0, pod(3n+2)≡ 2(-1)ⁿ⁺¹r₅(8n+5) \pmod9, and pod(5n+2)≡ 2(-1)ⁿr₃(8n+3) \pmod5. From which we deduce many interesting congruences including the following two infinite families of Ramanujan-type congruences: for a ∈ \11, 19\ and any integers α≥ 1 and n ≥ 0, we have pod(5^2α+2n+\fraca ⋅ 5^2α+1+18)≡ 0 \pmod5.

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