But a couple of weeks ago, he talked to me about something really intriguing. During a conference on high-performance computing, he told me about a number he had computed, and that even if he was not sure that the value was right, he considered it highly unlikely that anybody could prove him wrong, because there weren’t enough atoms in the universe to do so. How does that sound?
Well, it turns out that he was talking about the Skewes numbers. More specifically, he has computed the first Skewes number to be most probably equal to 1.397162914×10316. See another reference in Wolfram’s MathWorld.
You can find the complete presentation of that research here. The general idea is that we have something that is extremely difficult to compute, so we compute it at reduced precision, skipping through large ranges of numbers. In doing so, we do a probabilistic estimate of the risk that we miss the right values. It’s really low, to the point that finding a counter example would take more computations than what can have been done in the entire universe since the Big Bang. However, we clearly have not tested all possible values (which would take even more power).
OK, in reality, this depends on the behavior of the Zeta function, more specifically on the Riemann hypothesis. So there is still an infinitesimal chance that all this huge computing power went to waste, and that 1.397162914×10316 is actually not the first Skewes number. Time will tell…