Hello, I have found the local minimal point ( I can't explain why yet), Conjecture : there are no fractional part of positive Bernoulli numbers between 1/6 and 0.647247618323605748006387... That number 0.647247618323605748006387 is a local minimum which correspond to numbers of the form 207*10*1001*3^n that is {B(207*10*1001*3^n)} is minimal at 0.647247618323605748006387 and there are apparently no other values between 1/6 and that number. Why is it numbers of that form...I don't know. The big cluster of lines is at 1/6 of course since many B's have a fractional part of 1/6. I did the graph of {B(n)}, n=2,4,6,... to show how the fractional part behave for large n compared to a graph of succ. random lines, it is clearly not random at all. Simon Plouffe