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

**References**

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

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1999-05-26