A note on odd partition numbers (Q6564138)

For a positive integer \(n\) the partition function \(p(n)\) counts the non-increasing sequences of positive integers that sum to \(n\). This work is motivated by a result due to \textit{C.-S. Radu} [J. Reine Angew. Math. 672, 161--175 (2012; Zbl 1276.11165)] who proved the Subbarao's conjecture and showed that every arithmetic progression \(r\) (mod \(t\)) contains infinitely many integers \(N\equiv r\) (mod \(t\)) for which \(p(N)\) is even, and infinitely many integers \(M\equiv r\) (mod \(t\)) for which \(p(M)\) is odd. In this paper, the authors deal with partitions functions of the form \(p(lm+\delta_l)\), where \(0<\delta_l<l\).\N\NFrom the authors' abstract: ``For primes \(l\ge 5\), we give a new proof of the conclusion that there are infinitely many \(m\) for which \(p(lm+\delta_l)\) is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular forms congruences, which allows us to show that the smallest such \(m\) satisfies \(m<(l^2-1)/24\), representing a significant improvement to the previous bound.''