Vorobiev's Theorem

Vorobiev's theorem states that if F_l^2|F_k, then F_l|k, where F_n is a Fibonacci number and a|b means a divides b. The theorem was discovered by Vorobiev in 1942, but not published until 1967. It was used by Y. Matiyasevich in his negative solution to the Hilbert's tenth problem.


Note that the converse does not hold. For example, 2/F_3=1, but F_2/F_3^2=1/4. The plot above shows values of (k,l) for which F_l^2|F_k and F_l|k (black) and for which F_l^2F_k but F_l|k (red).

See also

Fibonacci Number, Hilbert's Problems

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Vorobiev, N. N. Fibonacci Numbers. Basel, Switzerland: Birkhäuser, 2002.

Referenced on Wolfram|Alpha

Vorobiev's Theorem

Cite this as:

Weisstein, Eric W. "Vorobiev's Theorem." From MathWorld--A Wolfram Web Resource.

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