Return to list of publications of J.P. Jones.

Jones, J.P., Diophantine representation of Fibonacci numbers over natural numbers Applications of Fibonacci Numbers vol. 3, Proceedings of the Third International Conference on Fibonacci Numbers and Applications, Pisa, Italy, July 25 -29, 1988, Kluwer Publishers, Dordrecht, 197-201.

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 shown that the set of Fibonacci numbers is singlefold diophantine definable in one unknown. In otherwords, that there exists a polynomial P(x,y) with parameter y and unknown x, such that for each y, y is a Fibonacci number if and only if (E!x)[P(x,y) = 0]. Here E!x means there exists a unique x.

From this result it follows that there is a polynomial Q(x,y), in two variables such that the set of Fibonacci numbers is identical with the set of nonnegative values of Q(x,y). Furthermore each Fibonacci number is taken on exactly once as a value of Q(x,y).

The following example of such a singlefold Fibonacci representing polynomial is given in the paper:

7y4x2 - 7y2x4 - 5yx5 + y3x3 + y5x - 2y6 + 3yx + 2y2 + 2y - x6 + x2 + x.


Return to list of publications of J.P. Jones.