Differences between perfect powers : prime power gaps

TL;DR

This paper develops advanced techniques combining Diophantine approximation, algebraic number theory, and modularity to explicitly determine prime power divisibility in differences of perfect powers, solving specific exponential equations.

Contribution

It introduces a comprehensive machinery that unifies various methods to analyze prime power divisibility in perfect powers, including solving specific exponential Diophantine equations.

Findings

Complete solution for the equation x^2 + q^α = y^n for primes q<100

New explicit criteria for prime power divisibility in perfect powers

Integration of multiple advanced mathematical techniques

Abstract

We develop machinery to explicitly determine, in many instances, when the difference $x^2-y^n$ is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and non-archimedean, lattice basis reduction, methods for solving Thue-Mahler and $S$-unit equations, and the Primitive Divisor Theorem of Bilu, Hanrot and Voutier) and classical Algebraic Number Theory, with results derived from the modularity of Galois representations attached to Frey-Hellegoaurch elliptic curves. By way of example, we completely solve the equation \[ x^2+q^\alpha = y^n, \] where $2 \leq q < 100$ is prime, and $x, y, \alpha$ and $n$ are integers with $n \geq 3$ and $\gcd (x,y)=1$.