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Squares from <i>D</i>(–4) and <i>D</i>(20) Triples

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Author(s): Zvonko Čerin

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 01;
Issue: 05;
Start page: 286;
Date: 2011;
Original page

Keywords: Fibonacci Numbers | Lucas Numbers | Square | Symmetric Sum | Alternating Sum | Product | Component

ABSTRACT
We study the eight infinite sequences of triples of natural numbers A=(F2n+1,4F2n+3,F2n+7), B=(F2n+1,4F2n+5,F2n+7), C=(F2n+1,5F2n+1,F2n+3), D=(F2n+3,4F2n+1,F2n+3) and A=(L2n+1,4L2n+3,L2n+7), B=(L2n+1,4L2n+5,L2n+7), C=(L2n+1,5L2n+1,L2n+3), D=(L2n+3,4L2n+1,L2n+3. The sequences A,B,C and D are built from the Fibonacci numbers Fn while the sequences A, B, C and D from the Lucas numbers Ln. 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.
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