Inspired by recent congruences by Andersen with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author for the function spt(n).… Expand

Let p(n) denote the partition function. In this article, we will show that congruences of the formexist for all primes m and l satisfying m≥13 and l≠2,3,m, where B is a suitably chosen integer… Expand

Proceedings of the National Academy of Sciences of the United States of America

2001

TLDR

It is reported that such congruences are much more widespread than was previously known, and the theoretical framework that appears to explain every known Ramanujan-type congruence is described.Expand

As has been known since the work of F.K.C. Rankin and H.P.F. Swinnerton-Dyer [8], the values of j(τ) at the zeros in H of the Eisenstein series Ek(τ) of any weight k on SL2(Z) always lie in the… Expand

AbstractLet p(n) denote the number of unrestricted partitions of a non-negative integer n. In 1919, Ramanujan proved that for every non-negative n
$$\begin{gathered} p(5 + 4) \equiv 0(\bmod 5),… Expand

seem to be distinguished by the fact that they are exceptionally rare. Recently, Ono [O1] has gone some way towards quantifying the latter assertion. More recently, Ono [O2] has made great progress… Expand

j(^r) = E c(n)xn = (1 +240 0 u3(n)xn) /xf24(x) 744, where 03(n) = d3. dln Then p(n) is just the number of unrestricted partitions of n, and c(n) is the Fourier coefficient of Klein's modular… Expand

We write and so that p(n) is the number of unrestricted partitions of n. Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24m ≡ 1 (mod qn), then p(m) ≡ 0 (mod qn). He proved his… Expand