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- W83201650 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." @default.
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- W83201650 date "1996-01-01" @default.
- W83201650 modified "2023-09-23" @default.
- W83201650 title "Congruences for partition functions" @default.
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- W83201650 doi "https://doi.org/10.1007/978-1-4612-4086-0_16" @default.
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