Congruence properties for the partition function

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Congruence properties for the partition function.

Eighty years ago, Ramanujan conjectured and proved some striking congruences for the partition function modulo powers of 5, 7, and 11. Until recently, only a handful of further such congruences were known. Here we report that such congruences are much more widespread than was previously known, and we describe the theoretical framework that appears to explain every known Ramanujan-type congruence.

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On Congruence Properties of the Partition Function

Some congruence properties of the partition function are proved.

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Congruence Properties of Binary Partition Functions

Let A be a finite subset of N containing 0, and let f(n) denote the number of ways to write n in the form ∑ j2 , where j ∈ A. We show that there exists a computable T = T (A) so that the sequence (f(n) mod 2) is periodic with period T . Variations and generalizations of this problem are also discussed.

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Generalized Congruence Properties of the Restricted Partition Function P (n,m)

Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for p(n,m), the closely related restricted partition function that enumerates the number of partitions of n into exactly m parts. Let ` be any odd prime. In this paper we establish explicit Ramanujan-type ...

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Congruence Properties modulo 5 and 7 for the Pod Function

In this paper, we prove arithmetic properties modulo 5 and 7 satisfied by the function pod(n) which denotes the number of partitions of n wherein odd parts must be distinct (and even parts are unrestricted). In particular, we prove the following: For all n ≥ 0, pod(135n + 8) ≡ 0 (mod 5), pod(135n + 107) ≡ 0 (mod 5), pod(135n + 116) ≡ 0 (mod 5), pod(675n + 647) ≡ 0 (mod 25), pod(3375n + 1997) ≡ ...

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ژورنال

عنوان ژورنال: Proceedings of the National Academy of Sciences

سال: 2001

ISSN: 0027-8424,1091-6490

DOI: 10.1073/pnas.191488598