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On Two Conjectures Concerning the Ternary Digits of Powers of Two
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Robert I. Saye

Lawrence Berkeley National Laboratory

Berkeley, CA 94720

USA

**Abstract:**

Erdős conjectured that 1, 4, and 256 are the only powers of two whose
ternary representations consist solely of `0`s and `1`s.
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* ≤ 2 · 3^{45} ≈ 5.9 × 10^{21}.
Our approach makes use of a simple recursive construction of
numbers 2^{n}
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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