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**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.**

The set of Fibonacci numbers *0,1,1,2,3,5,8,13,21,34,55,...* is
defined by the sequence *F _{0} = 0, F_{1} = 1 and
F_{n+2} = F_{n+1} + F_{n}*.

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:

*
*

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