Periodica Mathematica Hungarica

, Volume 64, Issue 1, pp 1–10

# On a family of diophantine triples {K,A2K + 2A, (A + 1)2K + 2(A + 1)} with two parameters II

• Bo He
• Alain Togbé
Article

## 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].

## Key words and phrases

Diophantine tuples simultaneous Diophantine equations

## Mathematics subject classification numbers

11D09 11D25 11J86

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