On the Diophantine system \(f(z)= f(x) f(y)= f(u) f(v)\)
The authors motivate this work by presenting a result of \textit{D. Chen} and \textit{T. Cai} [Rocky Mt. J. Math. 43, No. 1, 75--81 (2013; Zbl 1279.11029)] in the form: \[ \dbinom{z}{2} = \dbinom{x}{2}\dbinom{y}{2} = \dbinom{u}{2}\dbinom{v}{2} \] has infinitely many nontrivial positive integer solutions \((z, x, y, u, v)\). They then show that also \[ f (z) = f (x) f (y) = f (u) f (v) \] has infinitely many nontrivial solutions in positive integers or in the ring \(\mathbb Z[t]\) when \(f(X) = X^2-1\). A similar result for rational solutions when \(f(X) = X^2 +b\), is obtained using elementary theory of elliptic curves. See \textit{Sz. Tengely} and \textit{M. Ulas} [J. Number Theory 165, 67--83 (2016; Zbl 1373.11030)] for related results.
scientific article; zbMATH DE number 6850226
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