engleski

An interesting property of a recurrence related to the Fibonacci sequence

The sequence of Fibonacci numbers with even subscripts (F_{;2n};) has one remarkable property. If we choose three successive elements of this sequence, than the product of any two of them increased by 1 is a perfect square. Indeed, F_{;2n}; F_{;2n+2}; + 1 = F_{;2n+1};^2, F_{;2n}; F_{;2n+4}; + 1 = F_{;2n+2};^2. This property was studied and generalized by several authors. Hoggatt and Bergum proved that the number d = 4F_{;2n+1};F_{;2n+2};F_{;2n+3}; has the property that F_{;2n};d + 1, F_{;2n+2};d + 1 and F_{;2n+4};d + 1 are perfect squares, and Dujella proved that the positive integer d with the above property is unique. The purpose of this paper is to characterize linear binary recursive sequences which possess the similar property as the above property of Fibonacci numbers.

Fibonacci numbers; diophantine equations

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