• Corpus ID: 245329788

Congruences like Atkin's for the partition function

  title={Congruences like Atkin's for the partition function},
  author={Scott Ahlgren and Patrick Brodie Allen and Shiang Tang},
Let p(n) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p(Qln+ β) ≡ 0 (mod l) where l and Q are prime and 5 ≤ l ≤ 31; these lie in two natural families distinguished by the square class of 1− 24β (mod l). In recent decades much work has been done to understand congruences of the form p(Qmln + β) ≡ 0 (mod l). It is now known that there are many such congruences when m ≥ 4, that such congruences are scarce (if they exist at all) when m… 


Distribution of the partition function modulo composite integers M
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
Proof of a conjecture of Ramanujan
  • A. Atkin
  • Mathematics
    Glasgow Mathematical Journal
  • 1967
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
Distribution of the partition function modulo m
and he conjectured further such congruences modulo arbitrary powers of 5, 7, and 11. Although the work of A. O. L. Atkin and G. N. Watson settled these conjectures many years ago, the congruences
Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function
  • M. Newman
  • Mathematics
    Canadian Journal of Mathematics
  • 1957
If n is a non-negative integer, define p r(n) as the coefficient of x n in ; otherwise define p r(n) as 0. In a recent paper (2) the author established the following congruence: Let r = 4, 6, 8, 10,
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
Scarcity of congruences for the partition function
The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell
New Congruences for the Partition Function
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),
Arithmetic properties of the partition function
Let p(n) denote the number of partitions of the positive integer n; p(n) is the number of representations of n as a non-increasing sequence of positive integers (by convention, we agree that p(0) = 1
Congruence properties for the partition function
  • S. Ahlgren, K. Ono
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 2001
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.
Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences
Let p(n) denote the number of partitions of n. Let A,B ∈ N with A > B and ≥ 5 a prime, such that p(An+B) ≡ 0 (mod ), n ∈ N. Then we will prove that |A and ( 24B−1 ) = (−1 ) . This settles an open