 
Summary: A Note on the Number of Division Steps in the Euclidean Algorithm
S.A. Abramov
Computer Centre of the Russian Academy of Science,
Vavilova 40, Moscow 117967, Russia
abramov@ccas.ru
Let w be a natural number and let µ(w) be the maximal
number of divisions that the Euclidean algorithm,
a0 = q1a1 + a2 ,
a1 = q2a2 + a3 ,
· · · (1)
ak2 = qk1ak1 + ak ,
ak1 = qk ak ,
needs for a given input (a0 , a1 ), where a0 > a1 = w. Lam’e's
theorem [2, 1] (this theorem was proved earlier by Finck in
1841 [1]) implies the asymptotic estimate
µ(w) = O(log w), (2)
and log w cannot be replaced by any function h(w) such
that h(w) = o(log w), since, if F0 , F1 , . . . is the Fibonacci
sequence, for a0 = Fk+2 , w = a1 = Fk+1 the number of
divisions is equal to k. The di#erence between the latter
