Number Theory

2605 Submissions

[4] ai.viXra.org:2605.0064 [pdf] replaced on 2026-05-31 19:36:26

Fundamental NT-Theorem

Authors: Eugenio Evangelista Souza
Comments: 8 Pages. Math(Art) Version.

In this introduces a novel topological and algebraic framework designated as the Eugenio Numbers, establishing a rigorous mathematical mechanism to map coordinate trajectories from the complex plane into structured free monoids without geometric or informational degeneration. Traditional scalar representations of numeric systems intrinsically suffer from an irreversible loss of syntactic data, characteristically collapsing leading zeros and volatile word lengths. To resolve this fundamental limitation, we formalize the Factorized Floor Operator acting strictly upon the syntactic decomposition of structural expressions, anchoring the discrete projections of the complex domain via the newly defined Krishna Function. By equipping this sequential space with the non-Archimedean ultrametric of the Cantor topology, we prove the Fundamental Embedding Theorem, demonstrating that the analytical truncation error drives asymptotically to zero while completely preserving the spatial length and structural memory of digit blocks. Computational verification of the framework, including deterministic sequence indexing, is successfully implemented within the SageMath environment, opening new paradigms for lossless data compression, exact string indexing, and non-conventional numeration tracking. Synergy: Souza and Numerical Theorgyas (NT).
Category: Number Theory

[3] ai.viXra.org:2605.0033 [pdf] submitted on 2026-05-15 20:45:30

The Impossibility of the Specific Angle of Cosine Rule and Fermat’s Last Theorem

Authors: Md. Razib Talukder
Comments: 12 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)

This paper presents a self-contained proof of Fermat's Last Theorem, showing that for any exponent greater than two, no positive whole numbers satisfy the equation. The proof interprets any hypothetical solution geometrically as the sides of a triangle, applying the Law of Cosines to derive a strict condition involving an angle. This condition is expanded using the Binomial Theorem, revealing a combinatorial inequality that depends on the exponent. For exponents greater than two, this inequality contradicts the requirement that all three numbers be positive integers. A systematic case analysis of the possible ratios between the numbers resolves each possibility using elementary properties of binomial functions and basic inequalities. The entire proof avoids deep theories of elliptic curves and modular forms, using only geometry, algebra, and simple analytic reasoning. The key insight is that for exponents one and two, the geometric relationship holds perfectly, but for any larger exponent, it forces an impossible condition on the triangle's angle, breaking the requirement that all three sides be positive whole numbers.
Category: Number Theory

[2] ai.viXra.org:2605.0009 [pdf] submitted on 2026-05-05 05:23:36

Positivity of L'(x) Implies the Riemann Hypothesis

Authors: Yufei Liu
Comments: 5 Pages.

We prove that if L'(x)=∑_ρ (1/ρ) e^{-x/ρ} > 0 for all x>0, where the sum is taken in the symmetric pairing ρ,1-ρ, then all non-trivial zeros of the Riemann zeta function ζ(s) satisfy Re(ρ)=1/2. The proof uses the arithmetic representation of L'(x) derived in a previous paper, the Guinand--Weil explicit formula, and an asymptotic analysis of the prime sum. Assuming the existence of a zero with real part >1/2 leads to a contradiction by showing that L'(x) would become negative for a suitably chosen sequence of x.
Category: Number Theory

[1] ai.viXra.org:2605.0004 [pdf] submitted on 2026-05-04 01:27:59

Minimum Prime Gaps in Arithmetic Progressions: A Probabilistic Model Based on Hardy--Littlewood Correlations and Empirical Verification on 10^6 Primes

Authors: Giovanni Ferraiuolo
Comments: 6 Pages.

We propose a coherent probabilistic model for consecutive prime gaps inside a fixed arithmetic progression modulo M. The model combines a Cramér-type intensity filtered by the residue class, Hardy—Littlewood two-point correlations via the singular series S(g), and an exponential suppression of intermediate primes. Under natural assumptions, the relative frequencies of small admissible gaps satisfy freq(g2)/freq(g1) ~ S(g2)/S(g1). We test the model on the first 10^6 primes (up to 1.5×10^7) in the digital root classes modulo 9 using exact rational arithmetic in SageMath. The predicted resonance for gap 90 (excess factor 4/3) is observed as +34.5% against +33.3% predicted, an agreement within 1.2 percentage points over 166,567 gaps. For gaps approaching the mean spacing the two-body approximation breaks down, with a sign inversion at gap 198, clearly marking the transition scale. All computations confirm the structural Lemma 1 (zero violations) and the asymptotically stable product freq(g_min) × mean_gap ≈ 2 C2 S(g_min) φ(M).
Category: Number Theory