Repunit Coordinates for the Erdős Ternary-Digit Problem

Abstract

Paul Erdős conjectured that the only powers of two whose ternary expansions contain no digit 2 are 1, 4, and 256. Despite extensive computational verification, the conjecture remains open. We reformulate the problem using generalised base-4 repunits. Any admissible exponent is even, say \(n=2s\), and \[ 2^{2s}=4^s=3R_s(4)+1,\qquad R_s(4)=\frac{4^s-1}{3}. \] Thus \(2^{2s}\) omits the ternary digit 2 if and only if \(R_s(4)\) does. For each depth \(m\), the map \(Φ_m:s\bmod 3^m\mapsto R_s(4)\bmod 3^m\) is a permutation of the residue classes modulo \(3^m\). Hence every prescribed \(m\)-digit ternary suffix has a unique canonical exponent seed modulo \(3^m\), and all exponents producing that suffix form an arithmetic progression. The lifting identity \[ R_{r+q3^m}(4)\equiv R_r(4)+q3^m\pmod{3^{m+1}} \] gives three lifts at each depth, exactly two of which preserve admissibility. The finite maps assemble into an isometric homeomorphism of the 3-adic integers conjugating \(x \mapsto x+1\) with \(y \mapsto 4y+1\). The unresolved global integer problem remains. No proof of the Erdős conjecture is claimed.