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# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## Diophantine triples of Fibonacci numbers

### Tom 175 / 2016

Acta Arithmetica 175 (2016), 57-70 MSC: 11D09, 11D45, 11B37, 11J86. DOI: 10.4064/aa8259-6-2016 Opublikowany online: 15 September 2016

#### Streszczenie

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 $k=1$.

#### Autorzy

• Bo HeInstitute 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
e-mail
• Florian LucaSchool 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
e-mail
• Alain TogbéDepartment of Mathematics, Statistics, and Computer Science
Purdue University Northwest
1401 S. U.S. 421
Westville, IN 46391, U.S.A.
e-mail

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