On a family of Diophantine triples \(\{k,A^2k+2A,(A+1)^2k+2(A+1)\}\) with two parameters
Let \(A\), \(k\) and \(d\) be positive integers, and put \(H:=\{k,A^2k+2A,(A+1)^2k+2(A+1),d\}\). The authors prove that if \(H\) is a Diophantine quadruple, that is the product of any two distinct elements of \(H\) increased by \(1\) is a perfect square, then in case of \(3\leq A\leq 10\) \[ d=(4A^4+8A^3+4A^2)k^3+(16A^3+24A^2+8A)k^2+(20A^2+20A+4)k+(8A+4) \] holds. This result is an extension of a theorem of \textit{A. Dujella} [Publ. Math. 51, No. 3--4, 311--322 (1997; Zbl 0903.11010)] with \(A=1\), and also of a classical theorem of Baker and Davenport with \(A=k=1\).
scientific article; zbMATH DE number 5652037
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