What does this research mean for the field?

The OMARWA period P(m) of the integer sequence T_k = 6·2^k + 1 is defined as the minimal period of the sequence mod m, and is proven to be equal to ord_m'(2) where m' = m/gcd(m,6). Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to explore the modular periodic structure of the integer sequence Tk and establish a theory around it.

March 16, 2026Open Access

OMARWA Period Theory: Modular Period Structure and Fractal Laws of Tₖ = 6·2ᵏ + 1

Key Points

The aim is to explore the modular periodic structure of the integer sequence Tk and establish a theory around it.
Defined the OMARWA period P(m) for positive integers m.
Proved the Core Reduction Theorem connecting P(m) to ordm'(2).
Derived a super-period L = 30 through various approaches.
Verified principal theorems using Lean 4 software with 654 declarations.
Identified coincidences relating the modular period to exceptional Lie-theoretic Coxeter numbers.

Abstract

We study the integer sequence Tk = 6·2k + 1 (OEIS A004119) and establish a complete theory of its modular periodicity, introducing the OMARWA period P(m) framework. For each positive integer m, we define the OMARWA period P(m) as the minimal period of the sequence Tk mod m, and prove the Core Reduction Theorem: P(m) = ordm'(2) where m' = m/gcd(m,6). We derive fractal period laws, a super-period L = 30 via three independent paths, and observe coincidences with exceptional Lie-theoretic Coxeter numbers. All principal theorems are formally verified in Lean 4 with Mathlib (654 declarations, zero sorry axioms, 25 modules). Software DOI: 10.5281/zenodo.19024222.

OMARWA Period Theory: Modular Period Structure and Fractal Laws of Tₖ = 6·2ᵏ + 1

Key Points

Abstract

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