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## Turkish Journal of Mathematics

#### Article Title

Pythagorean triples containing generalized Lucas numbers

#### DOI

10.3906/mat-1702-102

#### Abstract

Let \$P\$ and \$Q\$ be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: \$U_{0}(P,Q)=0,U_{1}(P,Q)=1,\$ and \$ U_{n+1}(P,Q)=PU_{n}(P,Q)+QU_{n-1}(P,Q)\$ for \$n\geq 1\$ and \$ V_{0}(P,Q)=2,V_{1}(P,Q)=P,\$ and \$V_{n+1}(P,Q)=PV_{n}(P,Q)+QV_{n-1}(P,Q)\$ for \$n\geq 1,\$ respectively. In this paper, we assume that \$P\$ and \$Q\$ are relatively prime odd positive integers and \$P^{2}+4Q>0.\$ We determine all indices \$n\$ such that \$U_{n}=(P^{2}+4Q)x^{2}.\$ Moreover, we determine all indices \$n\$ such that \$(P^{2}+4Q)U_{n}=x^{2}.\$ As a result, we show that the equation \$V_{n}^{2}(P,1)+V_{n+1}^{2}(P,1)=x^{2}\$ has solution only for \$n=2,\$ \$P=1,\$ \$x=5\$ and \$V_{n+1}^{2}(P,-1)=V_{n}^{2}(P,-1)+x^{2}\$ has no solutions. Moreover, we solve some Diophantine equations.

#### Keywords

Generalized Fibonacci and Lucas numbers, Diophantine equations

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1912

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