Solving an infinite number of purely exponential Diophantine equations with four terms
Takafumi Miyazaki
TL;DR
The paper addresses the problem of solving $S$-unit-like Diophantine equations with at least four terms by focusing an infinite family of purely exponential equations with consecutive bases: $n^x+(n+1)^y+(n+2)^z=(n+3)^w$. The author combines elementary congruence arguments, Baker's method in both $p$-adic and rational forms, and the Bertók–Hajdu algorithm to fully solve the family for all $n \equiv 3 \pmod{4}$, showing that only $n=3$ yields solutions: $(x,y,z,w)=(3,1,1,2)$ and $(3,3,3,3)$. The work demonstrates how a hybrid approach can produce effective bounds in the multivariate exponential setting, reducing the problem to a finite check over eight small-base cases. This constitutes a first essentially complete resolution of infinitely many four-term exponential equations with consecutive bases and highlights the utility of combining congruence methods, $p$-adic and rational Baker bounds, and Skolem-type modular strategies in Diophantine problems. $k>3$ cases and the broader $S$-unit landscape are thereby advanced by providing a concrete, fully solved infinite family and a methodology transferable to related equations.
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 Bertók and Hajdu based on a variant of Skolem's conjecture on purely exponential equations.