Squares from <i>D</i>(–4) and <i>D</i>(20) Triples
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Abstract
We study the eight infinite sequences of triples of natural numbers <i>A</i>=(<i>F<sub>2n+1</sub></i>,4<i>F<sub>2n+3</sub></i>,<i>F<sub>2n+7</sub></i>), <i>B</i>=(<i>F<sub>2n+1</sub></i>,4<i>F<sub>2n+5</sub></i>,<i>F<sub>2n+7</sub></i>), <i>C</i>=(<i>F<sub>2n+1</sub></i>,5<i>F<sub>2n+1</sub></i>,<i>F<sub>2n+3</sub></i>), <i>D</i>=(<i>F<sub>2n+3</sub></i>,4<i>F<sub>2n+1</sub></i>,<i>F<sub>2n+3</sub></i>) and A=(<i>L<sub>2n+1</sub></i>,4<i>L<sub>2n+3</sub></i>,<i>L<sub>2n+7</sub></i>), B=(<i>L<sub>2n+1</sub></i>,4<i>L<sub>2n+5</sub></i>,<i>L<sub>2n+7</sub></i>), C=(<i>L<sub>2n+1</sub></i>,5<i>L<sub>2n+1</sub></i>,<i>L<sub>2n+3</sub></i>), D=(<i>L<sub>2n+3</sub></i>,4<i>L<sub>2n+1</sub></i>,<i>L<sub>2n+3</sub></i>. The sequences <i>A</i>,<i>B</i>,<i>C</i> and <i>D</i> are built from the Fibonacci numbers <i>F<sub>n</sub></i> while the sequences A, B, C and D from the Lucas numbers <i>L<sub>n</sub></i>. Each triple in the sequences <i>A</i>,<i>B</i>,<i>C</i> and <i>D</i> has the property <i>D</i>(-4) (<i>i. e</i>., 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 <i>D</i>(20). We show some interesting properties of these sequences that give various methods how to get squares from them.
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