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• 1 Neijiang Normal University Key Laboratory of Numerical Simulation of Sichuan Province Neijiang Sichuan 641112 P. R. China
• 2 Purdue University North Central Mathematics Department 1401 S, U.S. 421 Westville IN 46391 USA
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## Abstract

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

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\{ k,A^2 k + 2A,(A + 1)^2 k + 2(A + 1),d\}$$ \end{document}
. 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
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} d = (4A^4 + 8A^3 + 4A^2 )k^3 + (16A^3 + 24A^2 + 8A)k^2 \hfill \\ + (20A^2 + 20A + 4)k + (8A + 4) \hfill \\ \end{gathered}$$ \end{document}
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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## On relatively weakly almost Lindelöf subsets

Author: Yan-Kui Song