• Corpus ID: 219687128

Scarcity of congruences for the partition function

@article{Ahlgren2020ScarcityOC,
  title={Scarcity of congruences for the partition function},
  author={Scott Ahlgren and Olivia Beckwith and Martin Raum},
  journal={arXiv: Number Theory},
  year={2020}
}
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 $p(\ell n+\beta)\equiv 0\pmod\ell$ for the primes $\ell=5, 7, 11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+\beta)\equiv 0\pmod\ell$ where $Q\geq 5$ is prime and $m\geq… 
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