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A note on odd partition numbers

Michael Griffin, Ken Ono

Abstract

Ramanujan's celebrated partition congruences modulo $\ell\in \{5, 7, 11\}$ assert that $$ p(\ell n+δ_{\ell})\equiv 0\pmod{\ell}, $$ where $0<δ_{\ell}<\ell$ satisfies $24δ_{\ell}\equiv 1\pmod{\ell}.$ By proving Subbarao's Conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes $\ell \geq 5,$ we give a new proof of the conclusion that there are infinitely many $m$ for which $p(\ell m+δ_{\ell})$ is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such $m$ satisfies $m<(\ell^2-1)/24,$ representing a significant improvement to the previous bound.

A note on odd partition numbers

Abstract

Ramanujan's celebrated partition congruences modulo assert that where satisfies By proving Subbarao's Conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes we give a new proof of the conclusion that there are infinitely many for which is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such satisfies representing a significant improvement to the previous bound.
Paper Structure (6 sections, 5 theorems, 36 equations, 1 table)

This paper contains 6 sections, 5 theorems, 36 equations, 1 table.

Table of Contents

  1. Introduction and Statement of Results
  2. Preliminaries about modular form congruences
  3. A modulo $\ell$ Atkin-Lehner theorem
  4. Refinement of Sturm's Criterion
  5. Proof of Theorem \ref{['MainThm']}
  6. Proof of Theorem \ref{['MainThm']}

Key Result

Theorem 1.1

If $\ell \geq 5$ is prime, then there are infinitely many $m$ for which $p(\ell m+\delta_{\ell})$ is odd. Moreover, the smallest such $m$ satisfies $m<(\ell^2-1)/24.$

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 2.1: Theorem 1.1 of OnoRamsey
  • Theorem 2.2
  • Remark
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof