# 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},
journal={Transactions of the American Mathematical Society},
year={2013},
volume={365},
pages={4881-4894}
}
• 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.
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• Mathematics
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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
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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
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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