On two conjectures concerning the ternary digits of powers of two

Abstract

Erdős conjectured that 1, 4, and 256 are the only powers of two whose ternary representations consist solely of 0s and 1s. Sloane conjectured that, except for $\{2^0,2^1,2^2,2^3,2^4,2^{15}\}$, every other power of two has at least one 0 in its ternary representation. In this paper, numerical results are given in strong support of these conjectures. In particular, we verify both conjectures for all $2^n$ with $n\le 2\cdot 3^{45}\approx 5.9\times 10^{21}$. Our approach makes use of a simple recursive construction of numbers $2^n$ having prescribed patterns in their trailing ternary digits.

5 pages, 2 figures, 1 algorithm