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PreprintCC BY 4.0arXiv

Algebraic Proof of Completeness of a Positional Numeral System in the Normalized Mersenne Basis

  • E. Dyachenko
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Abstract

This paper introduces and investigates a non-classical positional numeral system constructed on the normalized Mersenne basis Mk = (2k−1)/3, structurally linking the classical Mersenne sequence (OEIS \textbf{A000225}) with the integer sequence of alternating binary bitmasks (OEIS \textbf{A002450}). The main objective of this study is to provide a rigorous algebraic proof of the system's completeness---specifically, theorems on the existence and absolute uniqueness of a canonical representation for any natural number. We show that the internal dynamics of the basis are deterministically governed by the splitting identity 3Mk = Mk+1 + 2Mk−1, which strictly restricts the valid alphabet of coefficients to the set {0, 1, 2}. Based on the lemma concerning the supremum of lower-order digits and a constructive greedy algorithm, it is proven that the proposed basis forms a closed and unambiguous algebraic space, entirely precluding the overlap of positional masses.

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Cite

BibTeXRIS
BibTeX
@misc{dyachenko2026algebraic,
  title = {{Algebraic Proof of Completeness of a Positional Numeral System in the Normalized Mersenne Basis}},
  author = {E. Dyachenko},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25091},
  eprint = {2609.25091},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25091},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Algebraic Proof of Completeness of a Positional Numeral System in the Normalized Mersenne Basis
AU  - E. Dyachenko
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.25091
UR  - https://arxiv.org/abs/2609.25091
AB  - This paper introduces and investigates a non-classical positional numeral system constructed on the normalized Mersenne basis M_k = (2^k−1)/3, structurally linking the classical Mersenne sequence (OEIS \textbf{A000225}) with the integer sequence of alternating binary bitmasks (OEIS \textbf{A002450}). The main objective of this study is to provide a rigorous algebraic proof of the system's completeness---specifically, theorems on the existence and absolute uniqueness of a canonical representation for any natural number. We show that the internal dynamics of the basis are deterministically governed by the splitting identity 3M_k = M_k+1 + 2M_k−1, which strictly restricts the valid alphabet of coefficients to the set {0, 1, 2}. Based on the lemma concerning the supremum of lower-order digits and a constructive greedy algorithm, it is proven that the proposed basis forms a closed and unambiguous algebraic space, entirely precluding the overlap of positional masses.
ER  -