On the Average Length of a Class of Finite Continued Fractions

Abstract

Many years ago Dr. J. Gillis asked me the following question: Let N and a be coprime natural integers, 1 ≦ a < N, so that a/N can be represented by a finite continued fraction

$$ a/N = 1/c_1 + 1/c_2 + \cdots + 1/c_{n(a)} $$

where the c _i are natural integers depending on N and a, and where c _n(a) > 1 (to make the representation unique).