The powers of two and three (Q809116)

One of the many questions of Erdős is the following: ``Does there exist an integer \(m\neq 0,2,8\) such that \(2^ m\) is a sum of distinct powers of 3?'' The author is able to show that the density of such m's is zero. In particular, if N(T) is the number of nonnegative integers \(m\leq T\) such that \(2^ m\) can be written as a sum of distinct powers of 3, then \(N(T)\leq 2.495 T^{\log 2/\log 3}\).