Congruences for partition functions

Abstract

The ordinary partition function p(n) and some of its generalizations satisfy some beautiful congruence properties. For instance, Ramanujan proved that for every integer n $$\begin{array}{*{20}c} {p(5n + 4) \equiv 0\,(\bmod \,5),} \\ {p(7n + 5) \equiv 0\,(\bmod \,7),} \\ {p(11n + 6) \equiv 0\,(\bmod \,11).} \\ \end{array}$$ Here we consider congruences for p(n) and c h (n), the number of partitions of n into h colors. If l is prime and s is a positive integer, then, using a result of Sturm, we compute a constant C(h, t, r, l S) such that c h (tn + r) ≡ 0 (mod l s) for all n if and only if the congruence holds for every n ≤ C(h, t, r, l s). If h = 1, these results pertain to p(n). In many cases, C(h, t, r, l S) is small enough that one obtains an effective method of determining the truth of alleged congruences. For example, Ramanujan’s congruences are easily verified because C(1, 5, 4, 5) = 2, C(1, 7, 5, 7) = 4, and C(1, 11,6,11) = 10.KeywordsPartition FunctionModular FormArithmetic ProgressionInteger CoefficientFinite Dimensional Vector SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.