# Congruences like Atkin's for the partition function

@inproceedings{Ahlgren2021CongruencesLA, title={Congruences like Atkin's for the partition function}, author={Scott Ahlgren and Patrick Brodie Allen and Shiang Tang}, year={2021} }

. Let p ( n ) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p ( Q 3 ℓn + β ) ≡ 0 (mod ℓ ) where ℓ and Q are prime and 5 ≤ ℓ ≤ 31; these lie in two natural families distinguished by the square class of 1 − 24 β (mod ℓ ). In recent decades much work has been done to understand congruences of the form p ( Q m ℓn + β ) ≡ 0 (mod ℓ ). It is now known that there are many such congruences when m ≥ 4, that such congruences are scarce (if…

## 2 Citations

### Scarcity of congruences for the partition function

- Mathematics
- 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…

### Congruence relations for r-colored partitions

- Mathematics
- 2022

Let l ≥ 5 be prime. For the partition function p(n) and 5 ≤ l ≤ 31, Atkin found a number of examples of primes Q ≥ 5 such that there exist congruences of the form p(lQn + β) ≡ 0 (mod l). Recently,…

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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…

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