On congruence properties of the 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.
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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 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 ...
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Throughout this paper, we shall use the following notation: c1,c2,. . . denote positive absolute constants. Z, N and N0 denote the set of integers, positive integers and non-negative integers respectively. The cardinality of a set S is denoted by |S|. bxc and {x} denote the integer part and the fractional part of x and ‖x‖ denotes the distance from x to the nearest integer: ‖x‖ = min({x}, 1 − {...
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ژورنال
عنوان ژورنال: International Journal of Mathematics and Mathematical Sciences
سال: 2000
ISSN: 0161-1712,1687-0425
DOI: 10.1155/s0161171200002829