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 representation. 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.