• 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(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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We write and so that p(n) is the number of unrestricted partitions of n. Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24m ≡ 1 (mod qn), then p(m) ≡ 0 (mod qn). He proved his
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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,
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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
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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 $p(\ell New Congruences for the Partition Function AbstractLet p(n) denote the number of unrestricted partitions of a non-negative integer n. In 1919, Ramanujan proved that for every non-negative n$\$\begin{gathered} p(5 + 4) \equiv 0(\bmod 5),
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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
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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