Congruences for the partition function in certain arithmetic progressions

Abstract

The partition function $\text{P(n)}$ satisfy some congruence properties. Eichhorn and Ono prove the existence of an effective constant $\text{C(m,r)}$ (where $\text{m,r∈}\text{N}$ have some restrictions), such that if $\text{p(mn+r)≡0(}\text{mod}\text{m)}$ for $\text{n⩽C(m,r)}$, then the congruence holds for every non-negative integer $\text{n}$. In this paper we improve the value of $\text{C(m,r)}$ by removing its dependence in $\text{r}$.