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

Radu, Cristian-Silviu

doi:10.1090/S0002-9947-2013-05777-7

Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences
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Trans. Amer. Math. Soc. 365 (2013), 4881-4894 Request permission

Abstract:

Let $p(n)$ denote the number of partitions of $n$. Let $A,B\in \mathbb {N}$ with $A>B$ and $\ell \geq 5$ a prime, such that \[ p(An+B)\equiv 0\pmod {\ell }, \quad n\in \mathbb {N}.\] Then we will prove that $\ell |A$ and $\left (\frac {24B-1}{\ell }\right ) \neq \left (\frac {-1}{\ell }\right )$. 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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