Certain Diophantine equations and new parity results for 21-regular partitions

2023/01/26 by Ajit Singh, Singh, Ajit, Gurinder Singh +3
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.11192

openalex publication_date 2023/01/26 · openalex created_date 2023/01/28 · openalex updated_date 2026/07/28

Abstract

For a positive integer t≥ 2, let b_t(n) denote the number of t-regular partitions of a nonnegative integer n. In a recent paper, Keith and Zanello investigated the parity of b_t(n) when t≤ 28. They discovered new infinite families of Ramanujan type congruences modulo 2 for b₂₁(n) involving every prime p with p≡ 13, 17, 19, 23 \pmod24. In this paper, we investigate the parity of b₂₁(n) involving the primes p with p≡ 1, 5, 7, 11 \pmod24. We prove new infinite families of Ramanujan type congruences modulo 2 for b₂₁(n) involving the odd primes p for which the Diophantine equation 8x²+27y²=jp has primitive solutions for some j∈\lbrace1,4,8\rbrace, and we also prove that the Dirichlet density of such primes is equal to 1/6. Recently, Yao provided new infinite families of congruences modulo 2 for b₃(n) and those congruences involve every prime p≥ 5 based on Newman's results. Following a similar approach, we prove new infinite families of congruences modulo 2 for b₂₁(n), and these congruences imply that b₂₁(n) is odd infinitely often.

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