Congruences for the partition function in certain arithmetic progressions (Q1969800)

\textit{D. Eichhorn} and \textit{K. Ono} [Prog. Math. 138, 309--321 (1996; Zbl 0852.11056)] showed that there is an effective constant \(C(m,r)\) such that the unrestricted partition function \(p(n)\) satisfies a congruence of the form \(p(mn+r) \equiv 0\pmod m\) for \(n\leq C(m,r)\). This paper improves the constant \(C(m,r)\) by removing its dependence on \(r\). The principal result states that if \(24r\equiv 1\pmod m\), where \(m\) is a prime \(\geq 5\), then \(p(mn+r) \equiv 0\pmod m\) for every nonnegative integer \(n\) if, and only if, the congruence holds for every \(n\leq(m^2-1)/24\).