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Nontrivial rational points on Erdős-Selfridge curves

Kyle Pratt

Abstract

We study rational points on the Erdős-Selfridge curves \begin{align*} y^\ell = x(x+1)\cdots (x+k-1), \end{align*} where $k,\ell\geq 2$ are integers. These curves contain "trivial" rational points $(x,y)$ with $y=0$, and a conjecture of Sander predicts for which pairs $(k,\ell)$ the curve contains "nontrivial" rational points where $y\neq 0$. Suppose $\ell \geq 5$ is a prime. We prove that if $k$ is sufficiently large and coprime to $\ell$, then the corresponding Erdős-Selfridge curve contains only trivial rational points. This proves many cases of Sander's conjecture that were previously unknown. The proof relies on combinatorial ideas going back to Erdős, as well as a novel "mass increment argument" that is loosely inspired by increment arguments in additive combinatorics. The mass increment argument uses as its main arithmetic input a quantitative version of Faltings's theorem on rational points on curves of genus at least two.

Nontrivial rational points on Erdős-Selfridge curves

Abstract

We study rational points on the Erdős-Selfridge curves \begin{align*} y^\ell = x(x+1)\cdots (x+k-1), \end{align*} where are integers. These curves contain "trivial" rational points with , and a conjecture of Sander predicts for which pairs the curve contains "nontrivial" rational points where . Suppose is a prime. We prove that if is sufficiently large and coprime to , then the corresponding Erdős-Selfridge curve contains only trivial rational points. This proves many cases of Sander's conjecture that were previously unknown. The proof relies on combinatorial ideas going back to Erdős, as well as a novel "mass increment argument" that is loosely inspired by increment arguments in additive combinatorics. The mass increment argument uses as its main arithmetic input a quantitative version of Faltings's theorem on rational points on curves of genus at least two.

Paper Structure

This paper contains 7 sections, 21 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline of the proofs
  4. Initial steps, and the key propositions
  5. Proofs of Propositions \ref{['prop:dense sequence has large gcds']}, \ref{['prop:not many ti are equal to one']}, and \ref{['prop:quant faltings theorem']}
  6. The mass increment argument
  7. The proof of Theorem \ref{['thm:ell = 3']}

Key Result

Theorem 1.1

Let $k$ be a sufficiently large positive integer, and let $\ell$ be a prime with If $\textup{gcd}(k,\ell)=1$, then any rational point $(x,y)\in \mathbb{Q}^2$ on the Erdős-Selfridge curve has $y=0$.

Theorems & Definitions (41)

  • Remark
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 4.1: Diophantine equation from nontrivial points
  • proof
  • Lemma 4.2: Factorization of terms
  • proof
  • Proposition 4.3: Many pairs with large GCD
  • Proposition 4.4: Few $t_i$ are trivial
  • ...and 31 more