Journal of Integer Sequences, Vol. 25 (2022), Article 22.3.4

On Two Conjectures Concerning the Ternary Digits of Powers of Two

Robert I. Saye
Lawrence Berkeley National Laboratory
Berkeley, CA 94720


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.

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(Concerned with sequences A351927 A351928.)

Received February 26 2022; revised versions received March 19 2022; March 21 2022. Published in Journal of Integer Sequences, March 23 2022.

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