Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer N, not of the form 10^k, such that the digits of N^2 are all 0s and 1s? It seems v …

Does anyone know of any lower bounds on the number of nonzero digits that appear in powers of 2 when written to base 3? (Other than the easy "If it's more than 8 it has to have at …