The Erd\H{o}s-Selfridge and the Schinzel-Tijdeman theorems hold in $PA^-$

TL;DR

This paper proves that certain number theory theorems, including the non-existence of powers among products of consecutive integers and results on polynomial solutions, hold within a weak logical framework called $PA^-$, which lacks induction.

Contribution

It demonstrates that key theorems in number theory remain valid in a weak, non-inductive fragment of Peano Arithmetic, expanding understanding of their logical strength.

Findings

The product of consecutive integers is never a perfect power.

Schinzel and Tijdeman's results on polynomial solutions hold in $PA^-$.

These theorems are valid without induction in discretely ordered rings.

Abstract

We show that "The product of consecutive integers is never a power" and several results by Schinzel and Tijdeman on the solutions of the equation $y^m=P(x)$, for $m>1$, $y>1$, and $P(x)$ a polynomial with rational coefficients and with at least two distinct zeros, hold in a weak fragment of Peano Arithmetic, $PA^-$, which lacks any kind of induction, and whose models are the positive cones of discretely ordered rings.