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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

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Abstract

Let A and k be positive integers. We study the Diophantine quadruples

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

. We prove that 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

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

when 3 ≦ A ≦ 10. This extends a theorem obtained by Dujella [7] for A = 1, and also, a classical theorem of Baker and Davenport [2] for A = k = 1.

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Authors and Affiliations

  1. Key Laboratory of Numerical Simulation of Sichuan Province, Neijiang Normal University, Neijiang, Sichuan, 641112, P. R. China

    B. He

  2. Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville, IN, 46391, USA

    A. Togbé

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  1. B. He
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  2. A. Togbé
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Correspondence to B. He.

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The first author is supported by the Foundation of Key Laboratory of Numerical Simulation of Sichuan Province. The second author is partially supported by Purdue University North Central.

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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. Acta Math Hung 124, 99–113 (2009). https://doi.org/10.1007/s10474-009-8155-5

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  • DOI: https://doi.org/10.1007/s10474-009-8155-5

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