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Jones, J.P., Diophantine representation of the Fibonacci numbers, Fibonacci Quarterly 13 (1975), 84-88. MR 52:3035.

Abstract

The set of Fibonacci numbers 0,1,1,2,3,5,8,13,21,34,55,... is defined by the sequence F0 = 0,   F1 = 1 and Fn+2 = Fn+1 + Fn.

In this paper it is proved that the set of Fibonacci numbers is identical with the set of positive values of a polynomial of the 5th degree in two variables x and y,
as these two variables range over the positive integers:

2y4x + y3x2 - 2y2x3 - y5 - yx4 + 2y.

Each Fibonacci number is a value of the above polynomial, for some positive integers x and y. Also each positive value of the polynomial is a Fibonacci number.


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