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Jones, J.P., Diophantine representation of the Lucas numbers, Fibonacci Quarterly 14 (1976), 134-134. MR 53:2818.

Abstract

The set of Lucas numbers 1,3,4,7,11,18,29,... is defined by the sequence Ln where L1 = 1,   L2 = 3 and Ln+2 = Ln+1 + Ln.

In this paper it is proved that the set of Lucas numbers is identical with the set of positive values of a polynomial of the 9th degree in two variables x and y,
as these two variables range over the positive integers:

y[1 - ((y2 - yx - x2)2 - 25)2].

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


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