# Ramanujan-type congruences modulo powers of 5 and 7

Ranganatha, D. (2017) Ramanujan-type congruences modulo powers of 5 and 7. Indian Journal of Pure and Applied Mathematics, 48 (3). pp. 449-465. ISSN 0975-7465

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Official URL: https://doi.org/10.1007/s13226-017-0237-4

## Abstract

Let bℓ(n) denote the number of ℓ-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of ℓ. In particular, they showed that for α, n ≥ 0, b25 (32α+3n+2 · 32α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where cN(n) counts the number of bipartitions (λ1,λ2) of n such that each part of λ2 is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, \$\${c_{25}}\backslashleft( {{5{2j}}n + \backslashfrac{{11 \backslashcdot {5{2j}} + 13}}{{12}}} \backslashright) \backslashequiv 0\$\$c25(52jn+11⋅52j+1312)≡0(mod 5j+1), \$\${c_{49}}\backslashleft( {{7{2j}}n + \backslashfrac{{11 \backslashcdot {7{_{2j}}} + 25}}{{12}}} \backslashright) \backslashequiv 0\$\$c49(72jn+11⋅72j+2512)≡0(mod 7j+1) and b25 (32α+3 · n+2 · 32α+2-1) ≡ 0 (mod 3 · 52j-1).

Item Type: Article Congruences, bipartitions, ℓ-regular partitions E Mathematical Science > Mathematics Department of > Mathematics C Swapna Library Assistant 01 Jun 2019 05:39 01 Jun 2019 05:39 http://eprints.uni-mysore.ac.in/id/eprint/2001