21 Dec
2002
21 Dec
'02
5:58 p.m.
Andreas Weingartner <weingartner@suu.edu> asks: << Prove or disprove: There exists an epsilon > 0, such that no natural number has the property that in base 2 as well as in base 3, [a fraction of at most epsilon] of the digits are nonzero.
I hereby conjecture that for any epsilon > 0 there is some 2^K and some 3^L such that either of them, expressed in either base 2 or base 3, has a fraction < delta of digits that are nonzero. [My intuition for this is partly based on the yet unproven -- but strongly supported by numerical evidence -- conjecture that the fractional parts of (3/2)^n, n = 1,2,3,..., are uniformly distributed in [0,1).] --Dan