Zhang, Yong; Chen, Deyi A Diophantine equation with the harmonic mean. (English) Zbl 1449.11070 Period. Math. Hung. 46, No. 1, 138-144 (2020). Fix a polynomial \(f(x) \in \mathbb{Q}[x]\). The authors consider the Diophantine equation for values \((x,y,z)\) defined by the harmonic mean of \(f(x), f(y)\) equalling \(f(z)\), that is, to the equation \(2 f(x) \, f(y) = f(z) \, (\,f(x) + f(y)\,)\). For \(f(x) = x^2 + bx + c\) with \(b, c\) integral and \(f\) without multiple roots, they show that should \((x, 2z + b, z)\) be an integer solution, then there are infinitely many integral solutions. Similarly, for a specific class of monic cubic \(f\) they give one solution for each \(f\). The techniques are Pell’s equation, elementary theory of elliptic curves, and clever algebraic manipulations. Reviewer: Thomas A. Schmidt (Corvallis) Cited in 1 Document MSC: 11D72 Diophantine equations in many variables 11D25 Cubic and quartic Diophantine equations 11D41 Higher degree equations; Fermat’s equation 11G05 Elliptic curves over global fields Keywords:Diophantine equation; Pell’s equation; integer solutions; rational parametric solutions PDFBibTeX XMLCite \textit{Y. Zhang} and \textit{D. Chen}, Period. Math. Hung. 46, No. 1, 138--144 (2020; Zbl 1449.11070) Full Text: DOI