# 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(Qln+ 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− 24β (mod l). In recent decades much work has been done to understand congruences of the form p(Qmln + β) ≡ 0 (mod l). It is now known that there are many such congruences when m ≥ 4, that such congruences are scarce (if they exist at all) when m…

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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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AbstractLet p(n) denote the number of unrestricted partitions of a non-negative integer n. In 1919, Ramanujan proved that for every non-negative n
$$\begin{gathered} p(5 + 4) \equiv 0(\bmod 5),…

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It is reported that such congruences are much more widespread than was previously known, and the theoretical framework that appears to explain every known Ramanujan-type congruence is described.

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Let p(n) denote the number of partitions of n. Let A,B ∈ N with A > B and ≥ 5 a prime, such that p(An+B) ≡ 0 (mod ), n ∈ N. Then we will prove that |A and ( 24B−1 ) = (−1 ) . This settles an open…