Title: Solving an infinite number of purely exponential Diophantine equations with four terms Show Chinese title

Title: 解含有四项的无限多个纯指数不定方程

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. Show Chinese abstract

Abstract: 在丢番图数论中，一个重要的未解问题是建立一种通用的方法，以有效找到任何给定的具有至少四个项的$S$-单位方程的所有解。 尽管文献中有很多工作致力于这个问题，其中大多数处理的是纯指数丢番图方程，但可以说，它们只在自然区分下解决了有限多个方程。 在本文中，我们研究了具有连续底数的四个项的无限多个纯指数丢番图方程。 我们的结果指出，方程$n^x+(n+1)^y+(n+2)^z=(n+3)^w$在正整数$n,x,y,z,w$中且$n \equiv 3 \pmod{4}$的解均由$(n,x,y,z,w)=(3,3,1,1,2), (3,3,3,3,3)$给出。 证明使用了在三元情况研究中发展出来的初等同余论证，在有理数和$p$-adic情况下使用的贝克方法，以及基于纯指数方程上的斯科伦猜想变体的伯特克和哈朱算法。