Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences
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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 \[ p(An+B)\equiv 0\pmod {\ell }, \quad n\in \mathbb {N}.\] 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.References
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Additional Information
- Cristian-Silviu Radu
- Affiliation: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, A-4040 Linz, Austria
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 4881-4894
- MSC (2010): Primary 11P83
- DOI: https://doi.org/10.1090/S0002-9947-2013-05777-7
- MathSciNet review: 3066773