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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A proof of the Hoggatt-Bergum conjecture


Author: Andrej Dujella
Journal: Proc. Amer. Math. Soc. 127 (1999), 1999-2005
MSC (1991): Primary 11D09, 11D25, 11B39; Secondary 11J86, 11Y50
Published electronically: March 17, 1999
MathSciNet review: 1605956
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Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that if $k$ and $d$ are positive integers such that the product of any two distinct elements of the set

\begin{displaymath}\{F_{2k},\, F_{2k+2},\, F_{2k+4},\, d\} \end{displaymath}

increased by $1$ is a perfect square, then $d$ has to be $4F_{2k+1}F_{2k+2}F_{2k+3}$. This is a generalization of the theorem of Baker and Davenport for $k=1$.


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Additional Information

Andrej Dujella
Affiliation: Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
Email: duje@math.hr

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04875-3
PII: S 0002-9939(99)04875-3
Keywords: Fibonacci numbers, property of Diophantus, simultaneous Pellian equations, linear form in logarithms, Baker-Davenport reduction procedure
Received by editor(s): October 23, 1997
Published electronically: March 17, 1999
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society