Computing Sciences

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 {20, 21, 22, 23, 24, 215}, every other power of two has at least one 0 in its ternary represen-tation. In this paper, numerical results are given in strong support of these conjectures. In particular, we verify both conjectures for all 2n with n ≤ 2 · 345 ≈ 5.9 × 1021. Our approach makes use of a simple recursive construction of numbers 2n having prescribed patterns in their trailing ternary digits.