Author(s)

Zvonko Čerin

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We study the eight infinite sequences of triples of natural numbers A=(F_2n+1,4F_2n+3,F_2n+7), B=(F_2n+1,4F_2n+5,F_2n+7), C=(F_2n+1,5F_2n+1,F_2n+3), D=(F_2n+3,4F_2n+1,F_2n+3) and A=(L_2n+1,4L_2n+3,L_2n+7), B=(L_2n+1,4L_2n+5,L_2n+7), C=(L_2n+1,5L_2n+1,L_2n+3), D=(L_2n+3,4L_2n+1,L_2n+3. The sequences A,B,C and D are built from the Fibonacci numbers F_n while the sequences A, B, C and D from the Lucas numbers L_n. Each triple in the sequences A,B,C and D has the property D(-4) (i. e., adding -4 to the product of any two different components of them is a square). Similarly, each triple in the sequences A, B, C and D has the property D(20). We show some interesting properties of these sequences that give various methods how to get squares from them.

Fibonacci Numbers, Lucas Numbers, Square, Symmetric Sum, Alternating Sum, Product, Component

Z. Čerin, "Squares from D(–4) and D(20) Triples," Advances in Pure Mathematics, Vol. 1 No. 5, 2011, pp. 286-294. doi: 10.4236/apm.2011.15052.