A Family of Cubic Diophantine Equations and 4-Chains

In a simple integer chain, if $u_{i-1}$, $u_i$, and $u_{i+1}$ are three consecutive terms of the chain, and the pair $(u_{i-1}, u_i)$ has a certain property, then the next pair $(u_i, u_{i+1})$ also has the same property. We extend the idea of a simple chain to an $n$-chain in which $n$ is a positive integer and if a pair $(u_{i-1}, u_{i})$ has a certain property, then the $n$th next pair $(u_{\pm n+i-1}, u_{\pm n+i})$ also has the same property. In this case, we call $(u_{i-1}, u_{i}, u_{i+1})$ and $(u_{\pm n+i-1}, u_{\pm n+i}, u_{\pm n+i+1})$ matching triples. We use $4$-chains to study a family of cubic Diophantine equations including $x^3 + y^3 + x +y +1 = xyz$ and three others. We show that a pair of integers $(x, y)$ satisfies one of those four equations if and only if $x$ and $y$ are consecutive terms of a $4$-chain. Our main result is that if triple $(u, t, vw)$ is an ordered list of three consecutive terms of one $4$-chain, where $|t|$ is a prime, $t$ does not divide $(u-v)$, and triple $(v, t, uw)$ is that of a second $4$-chain and it matches the first triple, then triple $(-w, t, -uv)$ is that of a third $4$-chain and it matches the other two triples.