2025/09/22 by Gedela Kavya Keerthana, Keerthana, G. Kavya, S. Ananya +3
Mathematics · Computer Science · #Advanced Mathematical Identities #Analytic Number Theory Research #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2509.17705
Let pk(n) denote the number of overpartition k-tuples of n. In 2023, Saikia \citesaikia conjectured the following congruences: pq(8n+2)amp; ≡ 0 \pmod4, pq(8n+3)≡ 0 \pmod8, pq(8n+4) ≡ 0 \pmod2,
pq(8n+5)amp; ≡ 0 \pmod8, pq(8n+6) ≡ 0 \pmod8, pq(8n+7)≡ 0 \pmod32, where n≥0 and q is prime. Recently, Sellers \citesellers2024elementary showed that these congruences hold for all odd integers q (not necessarily prime). In this paper, we show that the above congruences hold for all positive integers q (not necessarily odd). We also prove the following congruences on OPTk(n), the number of overpartition k-tuples with odd parts of n: For all i,j≥ 1, n≥ 0, r not a multiple of 2, k not a multiple of 2 or 3, and ℓ not a power of 2, nor a multiple of 2 or 3, we have OPT2i⋅ r(8n+7)amp; ≡ 0 \pmod2i+4, OPT3i⋅ 2j⋅ k(3n+2)amp; ≡ 0 \pmod3i+1⋅ 2j+2, OPT3i⋅ 2j⋅ k(3n+1)amp; ≡ 0 \pmod3i⋅ 2j+1,
OPT3i⋅ ℓ(3n+2)amp; ≡ 0 \pmod3i+1⋅ 2, OPT3i⋅ ℓ(3n+1)amp; ≡ 0 \pmod3i⋅ 2, where the first congruence was posed as a conjecture by Sarma et al. \citesaikiasarma and the latter four were conjectured by Das et al. \citeDSS.