On the Number of Divisions of the Euclidean Algorithm Applied to Gaussian Integers

• Authors: Heinrich Rolletschek
• Venue: Journal of Symbolic Computation
• Year: 1986

Two results about the Euclidean algorithm (EA) for Gaussian integers are proven in this paper: first, a general kind of division with remainder for Gaussian integers @x, @h is considered: @x = @c@h + @r, where we only require that @c is a Gaussian integer; N(@r), the norm of @r, need not be minimal or smaller than N(@h). This leads to a general version of the (EA), where an arbitrary remainder in this sense may be chosen at every division. We show that for every input the number of divisions is minimal, if a remainder of minimal norm is chosen at every step. We call such a version a minimal remainder-version of (EA). We also show that even the slightest deviation from a minimal remainder-version (in a sense to be defined) can lead to an increase in the number of divisions. Second, we establish a tight upper bound for the number of divisions of (EA) for an input of given size, and, equivalently, we determine the input values u, v with norm of u minimal, such that (EA) requires a given number of divisions. This is analogous to Lame's result for rational integers.