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On $$(\ell , m)$$ ( ℓ , m ) -regular partitions with distinct parts

Research paper by M. Prasad, K. V. Prasad

Indexed on: 03 May '18Published on: 01 May '18Published in: The Ramanujan Journal



Abstract

Let \(a_{\ell ,m}(n)\) denote the number of \((\ell ,m)\) -regular partitions of a positive integer n into distinct parts, where \(\ell \) and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for \(a_{3,5}(n)\) . For example, $$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$ where \(\alpha , \beta \ge 0\) .