firstname.lastname@example.org (Daniel McLaury) wrote: > An example of the "law of small numbers" in real maths > is given by the conjecture that Li(x) is always less than Pi(x). > This is true for values well over a googol, but eventually it > becomes false for all values over a certain number.
Slight correction: Littlewood actually showed that Li(x) - Pi(x) alternates in sign infinitely often, and gave an upper bound for the position of the first sign change.
Another example is Borsuk's conjecture on decomposing a bounded set in d dimensions into d+1 sets of strictly smaller diameter. It was shown to be false in dimension 1325 (and it's now known to be false in at least dimensions 561 and up).