## Diophantine representation of the Fibonacci numbers.(English)Zbl 0301.10010

Let $$F_n$$ be the $$n$$-th term in the Fibonacci sequence, defined by $$F_1 =1$$, $$F_2 =1$$, $$F_{n+2}=F_{n+1}+F_n$$. The author proves that the set of Fibonacci numbers is identical with the set of positive values of the polynomial $2y^4x + y^3x^2 - 2y^2x^3 - y^5 - yx^4 + 2y. \tag{1}$
The proof depends upon the fact that pairs of adjacent Fibonacci numbers, and only these, are to be found among the points with integer coordinates on the hyperbolas $$y^2 - yx - x^2 = \pm 1$$. Further, the author shows that the set of Fibonacci numbers is not the range of any polynomial and that his result above is best possible in the sense that the number of variables in (1) cannot be further decreased. He concludes by pointing out some connections between his result and Matijasevič’s solution of Hilbert’s Tenth Problem.

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11D41 Higher degree equations; Fermat’s equation
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### Online Encyclopedia of Integer Sequences:

Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.