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ON THE FAMILY OF DIOPHANTINE TRIPLES {k − 1, k + 1, 16k3 − 4k}

Published online by Cambridge University Press:  09 August 2007

YANN BUGEAUD ,
ANDREJ DUJELLA  and
MAURICE MIGNOTTE
YANN BUGEAUD
Affiliation:
Université Louis Pasteur, U. F. R. de Mathématiques, 7, rue René Descartes, 67084 Strasbourg, France e-mail: bugeaud@math.u-strasbg.fr
ANDREJ DUJELLA
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia e-mail: duje@math.hr
MAURICE MIGNOTTE
Affiliation:
Université Louis Pasteur, U. F. R. de Mathématiques, 7, rue René Descartes, 67084 Strasbourg, France e-mail: mignotte@math.u-strasbg.fr
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Abstract

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It is proven that if k ≥ 2 is an integer and 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 d = 4k or d = 64k5−48k3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k − 1, k + 1, c, d} are regular.

Keywords

11D0911D2511J8611Y50
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 2 , May 2007 , pp. 333 - 344
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

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