A simple proof of the Atkin-O'Brien partition congruence conjecture for powers of 13

TL;DR

The paper proves the Atkin–O'Brien partition congruence conjecture for powers of $13$ by embedding the problem in the theory of modular forms and eta-quotients. It combines Newman’s pole-order analysis for Fricke-invariant eta-quotients with the action of Hecke operators $T_{p^2}$ and the Atkin $U_p$ on carefully chosen eta-quotients, and then carries out a detailed $13$-adic valuation (adic) induction. The main result asserts that for all $\alpha\ge 1$ and primes $p\ge 5$, $p\neq 13$, there exists a constant $k=k(p,\alpha)$ such that $P(p^2 13^{\alpha} N) - \left\{ k - \left(\frac{-3 \cdot 13^{\alpha} N}{p}\right)p^{-2} \right\} P(13^{\alpha} N) + p^{-3} P\left(\frac{13^{\alpha} N}{p^2}\right) \equiv 0 \pmod{13^{\alpha}}$. This extends earlier cases ($\alpha=1,2$) to all $\alpha$, providing a streamlined modular-forms proof of the conjecture. The work clarifies how eta-quotients and Hecke actions govern partition congruences modulo $13^{\alpha}$ and offers a framework for similar results with other primes.

Abstract

In 1967, Atkin and O'Brien conjectured congruences for the partition function involving Hecke operators modulo powers of 13. In this paper, we provide a simple proof of this conjecture.

Theorems & Definitions (14)

• Theorem 1.1
• Conjecture 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Lemma 3.1: Newman
• Theorem 3.2: Newman
• Lemma 4.1: Atkin - O'Brien
• Lemma 4.2