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On a family of diophantine triples {K,A 2 K + 2A, (A + 1)2 K + 2(A + 1)} with two parameters II

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Abstract

Let A and k be positive integers. In this paper, we study the Diophantine quadruples

$$\{ k,A^2 k + 2A,(A + 1)^2 k + 2(A + 1)d\} .$$

If d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then

$$\begin{gathered} d = (4A^4 + 8A^3 + 4A^2 )k^3 + (16A^3 + 24A^2 + 8A)k^2 + \hfill \\ + (20A^2 + 20A + 4)k + (8A + 4) \hfill \\ \end{gathered} $$

for A ≥ 52330 and any k. This extends our result obtained in [4].

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Correspondence to Bo He.

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Communicated by Attila Pethő

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He, B., Togbé, A. On a family of diophantine triples {K,A 2 K + 2A, (A + 1)2 K + 2(A + 1)} with two parameters II. Period Math Hung 64, 1–10 (2012). https://doi.org/10.1007/s10998-012-9001-z

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  • DOI: https://doi.org/10.1007/s10998-012-9001-z

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