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


Author: Cristian-Silviu Radu
Journal: Trans. Amer. Math. Soc. 365 (2013), 4881-4894
MSC (2010): Primary 11P83
Published electronically: March 19, 2013
MathSciNet review: 3066773
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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

$\displaystyle p(An+B)\equiv 0\pmod {\ell }, \quad n\in \mathbb{N}.$

Then we will prove that $ \ell \vert 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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Additional Information

Cristian-Silviu Radu
Affiliation: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, A-4040 Linz, Austria

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05777-7
Keywords: Partition congruences of Ramanujan type, Ahlgren, Ono conjecture
Received by editor(s): February 24, 2011
Received by editor(s) in revised form: June 16, 2011, and December 7, 2011
Published electronically: March 19, 2013
Additional Notes: The author was supported by DK grant W1214-DK6 of the Austrian Science Funds FWF
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.