1Institute of Mathematics Aba Teachers University Wenchuan, Sichuan, 623000, P.R. China and Department of Mathematics Hubei University for Nationalities Enshi, Hubei, 445000, P.R. China 2School of Mathematics University of the Witwatersrand Private Bag X3, Wits 2050, South Africa and Centro de Ciencias Matemáticas UNAM Ap. Postal 61-3 (Xangari), CP 58089 Morelia, Michoacán, México 3Department of Mathematics, Statistics, and Computer Science Purdue University Northwest 1401 S. U.S. 421 Westville, IN 46391, U.S.A.
Acta Arithmetica, Tome 175 (2016) no. 1, pp. 57-70
Let $F_m$ be the $m$th Fibonacci number. We prove that if $F_{2n}F_k+1$ and $F_{2n+2}F_k+1$ are both perfect squares, then $k=2n+4$ for $n\ge 1$, or $k=2n-2$ for $n\ge 2$, except when $n=2$, in which case we can additionally have $k=1$.
Keywords:
mth fibonacci number prove perfect squares n except which additionally have
Affiliations des auteurs :
Bo He
1
;
Florian Luca
2
;
Alain Togbé
3
1
Institute of Mathematics Aba Teachers University Wenchuan, Sichuan, 623000, P.R. China and Department of Mathematics Hubei University for Nationalities Enshi, Hubei, 445000, P.R. China
2
School of Mathematics University of the Witwatersrand Private Bag X3, Wits 2050, South Africa and Centro de Ciencias Matemáticas UNAM Ap. Postal 61-3 (Xangari), CP 58089 Morelia, Michoacán, México
3
Department of Mathematics, Statistics, and Computer Science Purdue University Northwest 1401 S. U.S. 421 Westville, IN 46391, U.S.A.
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author = {Bo He and Florian Luca and Alain Togb\'e},
title = {Diophantine triples of {Fibonacci} numbers},
journal = {Acta Arithmetica},
pages = {57--70},
year = {2016},
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doi = {10.4064/aa8259-6-2016},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa8259-6-2016/}
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Bo He; Florian Luca; Alain Togbé. Diophantine triples of Fibonacci numbers. Acta Arithmetica, Tome 175 (2016) no. 1, pp. 57-70. doi: 10.4064/aa8259-6-2016
