Solving an infinite number of purely exponential Diophantine equations with four terms

Abstract

An important unsolved problem in Diophantine number theory is to establish a general method to effectively find all solutions to any given $S$-unit equation with at least four terms. Although there are many works contributing to this problem in literature, most of which handle purely exponential Diophantine equations, it can be said that all of them only solve finitely many equations in a natural distinction. In this paper, we study infinitely many purely exponential Diophantine equations with four terms of consecutive bases. Our result states that all solutions to the equation $n^x+(n+1)^y+(n+2)^z=(n+3)^w$ in positive integers $n,x,y,z,w$ with $n \equiv 3 \pmod{4}$ are given by $(n,x,y,z,w)=(3,3,1,1,2), (3,3,3,3,3)$. The proof uses elementary congruence arguments developed in the study of ternary case, Baker's method in both rational and $p$-adic cases, and the algorithm of Bert\'ok and Hajdu based on a variant of Skolem's conjecture on purely exponential equations.