Geodesic
Diophantine triples of Fibonacci numbers
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.
Acta Arithmetica, Tome 175 (2016) no. 1, pp. 57-70 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 $k=1$.
DOI : 10.4064/aa8259-6-2016
Keywords: mth fibonacci number prove perfect squares n except which additionally have

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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     title = {Diophantine triples of {Fibonacci} numbers},
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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

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