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, R_s(4)=(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:sbmod 3^m↦ 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)≡ R_r(4)+q3^mpmod3^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 ↦ x+1 with y ↦ 4y+1. The unresolved global integer problem remains. No proof of the Erdős conjecture is claimed.

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BibTeX RIS

BibTeX

@misc{idowu2026repunit,
  title = {{Repunit Coordinates for the Erdős Ternary-Digit Problem}},
  author = {Michael A. Idowu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03789},
  eprint = {2610.03789},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03789},
}

RIS

TY  - UNPB
TI  - Repunit Coordinates for the Erdős Ternary-Digit Problem
AU  - Michael A. Idowu
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.03789
UR  - https://arxiv.org/abs/2610.03789
AB  - 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, R_s(4)=(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:sbmod 3^m↦ 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)≡ R_r(4)+q3^mpmod3^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 ↦ x+1 with y ↦ 4y+1. The unresolved global integer problem remains. No proof of the Erdős conjecture is claimed.
ER  -

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