Lamé's Theorem

If $a$ is the smallest Integer for which there is a smaller Integer $b$ such that $a$ and $b$ generate a Euclidean Algorithm remainder sequence with $n$ steps, then $a$ is the Fibonacci Number $F_{n+2}$. Furthermore, the number of steps in the Euclidean Algorithm never exceeds 5 times the number of digits in the smaller number.

See also Euclidean Algorithm

References

Honsberger, R. ``A Theorem of Gabriel Lamé.'' Ch. 7 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 54-57, 1976.