A real number x is normal to base b if all finite words w over the alphabet {0,1, ..., b-1} occur as factors of the base-b expansion of x with a limiting frequency equal to b^-|w|. A number is normal if it is normal in all integer bases b ≥ 2. Although there are some numbers, such as Chaitin's ω, which are known to be normal to all bases, and although it is known that almost all real numbers are normal to all bases, nothing is known about the "classical" numbers such as π, e, √2 and log 2.

Even much weaker questions, such as whether any particular factor occurs infinitely often in π, e, √2 and log 2, have no answers currently (except trivial ones, such as whether π contains infinitely many 1's in its base-2 expansion).

-- JeffreyShallit - 13 Oct 2010