What does this research mean for the field?

The integer sequence T_k = 6*2^k + 1 exhibits modular period structures and fractal laws that reveal algebraic universality. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to explore the modular period structure and algebraic properties of the affine sequence T_k.

March 17, 20260 citationsOpen Access

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Modular Period Structure, Fractal Laws, and Algebraic Universality of the Affine Sequence Tₖ = 6*2ᵏ + 1

OCOmer Cetintas

Key Points

The aim is to explore the modular period structure and algebraic properties of the affine sequence T_k.
Utilized 28 modules of OmarwaCore Lean 4 formalization.
Applied the Core Reduction Theorem P(m).
Investigated Fractal Period Laws and Product Theorem via CRT.
Derived the Super-Period L and explored its implications.
Developed connections between various mathematical functions and structures.
Identified a total of 956 theorems related to the sequence.
Demonstrated the GCD Law's role in sequence analysis.
Established relationships using palindromic involution and bridge constructions.

Abstract

A comprehensive number-theoretic study of the integer sequence Tₖ = 6*2ᵏ + 1 (OEIS A004119). All 28 modules of the OmarwaCore Lean 4 formalization are covered: Core Reduction Theorem P (m) = ord (2, m/gcd (m, 6) ), Fractal Period Laws P (pⁿ) = d₀ * p^ (n-1), Product Theorem via CRT, Super-Period L = 30 with three independent derivations, GCD Law, Palindromic Involution, Pascal-Sierpinski-Fibonacci bridge, and Grand Synthesis five-way theorem. 956 theorems, 0 sorry axioms. 13 pages.

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Modular Period Structure, Fractal Laws, and Algebraic Universality of the Affine Sequence Tₖ = 6*2ᵏ + 1

Key Points

Abstract

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