In his original paper Brent proposed several ways to approach the determination of the magic constant b . That week of July 14th, Knuth and Brent were working in parallel with two different methods to estimate theses constants. Now, something strange kept happening. The two methods would produce values in agreement for about 12 digits, but never more. All this within hours of the time when Knuth in California and Brent in Australia were starting to suspect that something was going wrong with the two methods.
Nobody seems to have full answers yet.
Indeed, periodic fluctuations should be there. Very roughly, the binary gcd involves harmonic sums with frequencies that are powers of two. Such fluctuations are often small, easily passing unnoticed. Brigitte has an analysis conditioned upon a few highly plausible conjectures that agrees very well with numerical computations of Richard and Don while explaining otherwise unsuspected relations between various structure constants of the binary gcd algorithm. Many questions remain, and we hope that a complete analysis will be available in the near future.