PORTUGALIAE MATHEMATICA Vol. 52, No. 3, pp. 305318 (1995) 

Diophantine Quadruples for Squares of Fibonacci and Lucas NumbersAndrej DujellaDepartment of Mathematics, University of Zagreb,Bijenicka cesta 30, 41000 ZAGREB  CROATIA 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 $$ \Bigl\{2F_{n1},\,2F_{n+1},\,2F_{n}^{3}F_{n+1}F_{n+2},\, 2F_{n+1}F_{n+2}F_{n+3}(2F_{n+1}^{2}F_{n}^{2})\Bigr\} $$ has the property $D(F_{n}^{2})$ and that the sets $$ \eqalign{&{}\Bigl\{2F_{n2},\,2F_{n+2},\,2F_{n}L_{n1}L_{n}^{2}L_{n+1}, \,10F_{n}L_{n1}L_{n+1}[L_{n1}L_{n+1}(1)^{n}]\Bigr\},\cr &{}\Bigl\{F_{n3}F_{n2}F_{n+1},\,F_{n1}F_{n+2}F_{n+3},\,F_{n}L_{n}^{2}, \,4F_{n1}^{2}F_{n}F_{n+1}^{2}(2F_{n1}F_{n+1}F_{n}^{2})\Bigr\}\cr} $$ have the property $D(L_{n}^{2})$. Keywords: Fibonacci numbers; Lucas numbers; property of Diophantus. Classification (MSC2000): 11B39, 11D09 Full text of the article:
Electronic version published on: 29 Mar 2001. This page was last modified: 27 Nov 2007.
© 1995 Sociedade Portuguesa de Matemática
