Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences

@article{Radu2013ProofOA,
  title={Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences},
  author={Cristian-Silviu Radu},
  journal={Transactions of the American Mathematical Society},
  year={2013},
  volume={365},
  pages={4881-4894}
}
  • Cristian-Silviu Radu
  • Published 19 March 2013
  • Mathematics
  • Transactions of the American Mathematical Society
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 problem by Scott Ahlgren and Ken Ono. Our proof is based on results by Deligne and Rapoport. 
Congruences like Atkin's for the partition function
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
Congruences on Square-Classes for the Partition Function
We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in
Arithmetic properties of septic partition functions
Congruences and related identities are derived for a set of colored and weighted partition functions whose generating functions generate the graded algebra of integer weight modular forms of level
Nonholomorphic Ramanujan-type congruences for Hurwitz class numbers
TLDR
It is discovered that Ramanujan-type congruences for Hurwitz class numbers can be supported on nonholomorphic generating series and a divisibility result is established for such non holomorphic congruence of Hurwitzclass numbers.
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
Relations among Ramanujan-Type Congruences II
We show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra and provide several structure results for them. We discover a dichotomy between congruences
Relations among Ramanujan-Type Congruences I.
We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific Hecke congruences.More generally, for weakly
Congruences of Hurwitz class numbers on square classes
We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish
Relations among Ramanujan-Type Congruences I Ramanujan-type Congruences in Integral Weights
We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific Hecke congruences. More generally, for weakly
...
...

References

SHOWING 1-10 OF 24 REFERENCES
Distribution of the partition function modulo $m$
Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin,
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
Note on Partitions Modulo 5
In the paper preceding this one, Parkin and Shanks study the distribution of the values of the unrestricted partition function p(n) modulo 2 and come to the conclusion that there is no apparent
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
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.
SOME PROPERTIES OF p(n) AND c(n) MODULO POWERS OF 13
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
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
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
Dyson's crank of a partition
holds. He was thus led to conjecture the existence of some other partition statistic (which he called the crank); this unknown statistic should provide a combinatorial interpretation of ^-p(lln + 6)
Cranks andt-cores
SummaryNew statistics on partitions (calledcranks) are defined which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given
Modular forms on noncongruence subgroups
In this paper I will describe some results and open problems connected with noncongruence subgroups of PSL2(z). Most of these have their origins in the fundamental paper of Atkin and Swinnerton-Dyer
...
...