Handling some Diophantine equation via Euclidean algorithm and its application to purely exponential equations

Abstract

In this paper, we use a variety of classical and new research methods for ternary exponential Diophantine equations and extensive use of computer calculations to study the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for listed specific cases. Precisely, we confirm that for any fixed prime $c$ of the form $2^r \cdot 3 +1$ with some positive integer $r$ the conjecture holds true, except for only finitely many cases all of which can be effectively determined. Most importantly we prove the conjecture to be true whenever $c = 7, 13$, or $97$, giving another proof of the result of T. Miyazaki and I. Pink for $c=13$. We also contribute to the estimation of the number of positive integer solutions $(x,y)$ to the equation $a^x-b^y=c$ for any fixed positive integers $a,b$ and $c$ with both $a$ and $b$ greater than 1. Further, based on a key idea in the proofs of the above results, we present a new application of the Euclidean algorithm for polynomials to the polynomial-exponential Diophantine equation \[ X^m - X^n = q^{y_1} - q^{y_2} \] in positive integers $X, y_1$ and $y_2$, where $m$ and $n$ are given positive integers with $m>n$, and $q$ is a given prime.