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A proof of the Hoggatt-Bergum conjecture

Author(s): Andrej Dujella
Journal: Proc. Amer. Math. Soc. 127 (1999), 1999-2005.
MSC (1991): Primary 11D09, 11D25, 11B39; Secondary 11J86, 11Y50
Posted: March 17, 1999
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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, Bijenicka cesta 30, 10000 Zagreb, Croatia
Email: duje@math.hr

DOI: 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
Posted: March 17, 1999
Communicated by: David E. Rohrlich
Copyright of article: Copyright 1999, American Mathematical Society

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The following works have cited this article

A. Dujella, Complete solution of a family of simultaneous Pellian equations, Acta Math. Inform. Univ. Ostraviensis 6 (1998), 59-67. MR CMP 1 822 516

A. Dujella and A. Pethoe, Integer points on family of elliptic curves, Publ. Math. Debrecen 56 (2000), 321-335. MR 2001f:11087

A. Dujella, A parametric family of elliptic curves, Acta Arith. 94 (2000), 87-101. MR 2001c:11062

A. Dujella, Diophantine triples and construction of high-rank elliptic curves over Q with three nontrivial 2-torsion points, Rocky Mountain J. Math. 30 (2000), 157-164. MR 2001a:11090

A. Dujella, An absolute bound for the size of Diophantine m-tuples, J. Number Theory 89 (2001), 126-150. MR CMP 1 838 708

A. Dujella, Diophantine m-tuples and elliptic curves, J. Theor. Nombres Bordeaux 13 (2001), 111-124. MR CMP 1 838 074

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