Congruence properties of the m-ary partition function
نویسندگان
چکیده
منابع مشابه
On Congruence Properties of the Partition Function
Some congruence properties of the partition function are proved.
متن کاملCongruence 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.
متن کاملGeneralized 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 ...
متن کاملCongruence 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.
متن کاملOn m–ary Partition Function Congruences: A Fresh Look at a Past Problem
Let bm(n) denote the number of partitions of n into powers of m. Define σr = ε2m 2 + ε3m 3 + · · · + εrm, where εi = 0 or 1 for each i. Moreover, let cr = 1 if m is odd, and cr = 2 r−1 if m is even. The main goal of this paper is to prove the congruence bm(m n− σr −m) ≡ 0 (mod m/cr). For σr = 0, the existence of such a congruence was conjectured by R. F. Churchhouse some thirty years ago, and i...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1971
ISSN: 0022-314X
DOI: 10.1016/0022-314x(71)90051-5