Diophantine Quadruples for Squares of Fibonacci and Lucas Numbers

Andrej Dujella

Abstract: Let $n$ be an integer. A set of positive integers is said to have the property $D(n)$ if the product of its any two distinct elements increased by $n$ is a perfect square. In this paper, the sets of four numbers represented in terms of Fibonacci numbers with the property $D(F_{n}^{2})$ and $D(L_{n}^{2})$, where $(F_{n})$ is the Fibonacci sequence and $(L_{n})$ is the Lucas sequence, are constructed. Among other things, it is proved that the set $$ \Bigl\{2F_{n-1},\,2F_{n+1},\,2F_{n}^{3}F_{n+1}F_{n+2},\, 2F_{n+1}F_{n+2}F_{n+3}(2F_{n+1}^{2}-F_{n}^{2})\Bigr\} $$ has the property $D(F_{n}^{2})$ and that the sets $$ \eqalign{&{}\Bigl\{2F_{n-2},\,2F_{n+2},\,2F_{n}L_{n-1}L_{n}^{2}L_{n+1}, \,10F_{n}L_{n-1}L_{n+1}[L_{n-1}L_{n+1}-(-1)^{n}]\Bigr\},\cr &{}\Bigl\{F_{n-3}F_{n-2}F_{n+1},\,F_{n-1}F_{n+2}F_{n+3},\,F_{n}L_{n}^{2}, \,4F_{n-1}^{2}F_{n}F_{n+1}^{2}(2F_{n-1}F_{n+1}-F_{n}^{2})\Bigr\}\cr} $$ have the property $D(L_{n}^{2})$.

Keywords: Fibonacci numbers; Lucas numbers; property of Diophantus.

Classification (MSC2000): 11B39, 11D09

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