• Corpus ID: 245329788

# 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}
}
• Published 17 December 2021
• Mathematics
. 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
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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 ## References SHOWING 1-10 OF 35 REFERENCES Distribution of the partition function modulo composite integers M seem to be distinguished by the fact that they are exceptionally rare. Recently, Ono [O1] has gone some way towards quantifying the latter assertion. More recently, Ono [O2] has made great progress Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function • M. Newman • Mathematics Canadian Journal of Mathematics • 1957 If n is a non-negative integer, define p r(n) as the coefficient of x n in ; otherwise define p r(n) as 0. In a recent paper (2) the author established the following congruence: Let r = 4, 6, 8, 10, SOME PROPERTIES OF p(n) AND c(n) MODULO POWERS OF 13 • Mathematics • 1967 j(^r) = E c(n)xn = (1 +240 0 u3(n)xn) /xf24(x) 744, where 03(n) = d3. dln Then p(n) is just the number of unrestricted partitions of n, and c(n) is the Fourier coefficient of Klein's modular New Congruences for the Partition Function 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), Arithmetic properties of the partition function • Mathematics • 2003 Let p(n) denote the number of partitions of the positive integer n; p(n) is the number of representations of n as a non-increasing sequence of positive integers (by convention, we agree that p(0) = 1 Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences . Let p ( n ) denote the number of partitions of n . Let A,B ∈ N with A > B and (cid:2) ≥ 5 a prime, such that p ( An + B ) ≡ 0 (mod (cid:2) ) , n ∈ N . Then we will prove that (cid:2) | A and Modular Functions In Analytic Number Theory Knopp's engaging book presents an introduction to modular functions in number theory by concentrating on two modular functions,$\eta(\tau)$and$\vartheta(\tau)\$, and their applications to two
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