An Infinite Two-Parameter Family of Diophantine Triples

In this paper, we study sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. It is shown that if the two smallest elements have the form KA2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ KA}^2$$\end{document}, 4KA4±4A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4{ KA}^4 \pm 4 A$$\end{document} for some positive integers A and K, and the third one is chosen canonically, then any such set consisting of three elements can be contained in a unique such set with four elements.