2022/12/19 by Cristina Ballantine, Ballantine, Cristina, Mircea Merca +3
Mathematics · #05A17 #11D09 #11D45 #11F33 #11P83 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2212.09810
openalex publication_date 2022/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let b3(n) be the number of 3-regular partitions of n. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo 2 for b3(2n) involving every prime p with p ≡ 13, 17, 19, 23 \pmod 24, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo 2 for b3(2n) involving every prime p\geqslant 5. In this paper, we introduce new infinite Ramanujan type congruences modulo 2 for b3(2n). They complement naturally the results of Keith-Zanello and Yao and involve primes in \mathcal P=\p prime : ∃ j∈ \1,4,8\, x, y ∈ \mathbb Z, gcd(x,y)=1 \text with x2+216y2=jp\ whose Dirichlet density is 1/6. As a key ingredient in our proof we show that of the number of primitive solutions for x2+216y2=pm, p ∈ \mathcal P, p\nmid m and pm≡ 1\pmod24, is divisible by 8. Here, the difficulty arises from the fact that 216 is not idoneal. We also give a conjectural exact formula for the number of solutions for this Diophantine equation. In the second part of the article, we study reversals of Euler-type identities. These are motivated by recent work of the second author on a reversal of Schur's identity which involves 3-regular partitions weighted by the parity of their length.