On the Diophantine equation \((x^2+k)(y^2+k)=(z^2+k)^2\)
With fixed \(k\), a solution to the equation in the title gives three values of the polynomial \(x^2 +k\) forming a geometric progression. First the author reproves a result known since 1960's that for \(k\) of the shape \(\pm(a^2-2b^2)\) there are infinitely many solutions in integers \(x,y,z\). Then he proves that for infinitely many integers \(k\) there exists a four term geometric progression of integers of the form \(x^2 + k\). Reviewer's remark. A case of special interest is when \(k=-1\) and \(x,y,z\) are odd since then a solution to the equation produces three triangular numbers in a geometric progression. It was proved recently by \textit{Y. G. Chen} and \textit{J. H. Fang} [Integers 7, No. 1, Paper A19, 2 p. (2007; Zbl 1130.11014)] that there are no geometric progressions formed by four triangular numbers.
systems of quadratic Diophantine equations
scientific article; zbMATH DE number 5541312