On Jan 25, 8:53 pm, Dmitry Shintyakov <shintya...@gmail.com> wrote: > Everybody, thank you for your attention to my little conjecture. > > First of all, I am grateful to Matt and David Bernier for pointing out > that the conjecture is wrong. > Actually, it was very hard for me to believe it. I played with this > problem for a quite long time and the > perfect double-precision match between x_max and 36*pi/127 completely > convinced me that the match is exact. > > First I thought that there can be some software glitch and tried to > repeat your calculations for x = 36*Pi/127 - 1E-54, > using 2000-bit (600 decimal digits) floats (quite an overkill huh). > And I have found no mistake. > > After few minutes of calculation, I have made logarithmic graphs of > differential quotient at very small > scales of dx (from 1E-200 to 1E-20), and this became obvious. > Here they are:http://dmishin.blogspot.com/2009/01/36127-conjecture-failed.html > > This graph helped me to find better counter-example: > > x = 36*Pi/127 - 1.998e-29 > > This gives f(x) - f(36*Pi/127) \approx 3.06E-32. It seems to be best > approximation to x_max for now.
Yes, I belatedly ran my program again at a "higher resolution" and it came up with the same x-value as I posted elsewhere in this thread a while ago (and was subsequently extended by Robert Israel), namely
x = 0.8905302010175791857059461558702704369...
which differs from 36*pi/127 by, as you say, about 1.998E-29. My program is not absolutely guaranteed to find the global maximum, but based on my results I'm, let's say, about 75% confident that this really is the best that can be achieved (to the given precision).