On two conjectures concerning the ternary digits of powers of two

TL;DR

This paper provides strong numerical evidence supporting two conjectures about the ternary digit patterns of powers of two, verifying them for extremely large exponents using a recursive digit construction method.

Contribution

It introduces a recursive approach to analyze ternary digits of powers of two and verifies two longstanding conjectures up to very large exponents.

Findings

Confirmed Erd ext{"o}s' conjecture for n  2  2 imes 3^{45}

Supported Sloane's conjecture for all n 2 2 2 2 2^{15}

Developed a recursive method for constructing powers of two with specific ternary digit patterns

Abstract

Erd\H{o}s 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 $n \leq 2 \cdot 3^{45} \approx 5.9 \times 10^{21}$. Our approach makes use of a simple recursive construction of numbers $2^n$ having prescribed patterns in their trailing ternary digits.