Author: Ivan Soldo 1
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  • 1 University of Osijek Department of Mathematics Trg Ljudevita Gaja 6 31000 Osijek Croatia
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Let b = 2, 5, 10 or 17 and t > 0. We study the existence of D(−1)-quadruples of the form {1, b, c, d} in the ring \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} \(\mathbb{Z}\left[ {\sqrt { - t} } \right]\) \end{document}. We prove that if {1, b, c} is a D(−1)-triple in \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} \(\mathbb{Z}\left[ {\sqrt { - t} } \right]\) \end{document}, then c is an integer. As a consequence of this result, we show that for t ∉ {1, 4, 9, 16} there does not exist a subset of \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} \(\mathbb{Z}\left[ {\sqrt { - t} } \right]\) \end{document} of the form {1, b, c, d} with the property that the product of any two of its distinct elements diminished by 1 is a square of an element in \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} \(\mathbb{Z}\left[ {\sqrt { - t} } \right]\) \end{document}.

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