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Stratospheric Ozone Variability Linked to Dynamical Forcing in the Arctic Winter 2023/2024
Dora Pancheva
,Plamen Mukhtarov
Posted: 06 March 2026
Mechanical Fatigue of Titanium Dental Implants after Implantoplasty: Effect of Material Removal and Finite Element Simulations
Esteban Padullés-Roig
,Pablo Sevilla
,Eugenio Velasco-Ortega
,Miguel Cerrolaza
,Darcio Fonseca
,Jeanne Parache
,Conrado Aparicio
,Javier Gil
Posted: 06 March 2026
From the Pythagorean Dream to the Fermatian Obstruction: The Unified Chain
Ibar Federico Anderson
For every prime p and every integer a, the backward finite difference δp(a) := aᵖ − (a − 1)ᵖ equals the cyclotomic binary form Φp(a, a − 1) and hence the norm N_Q(ζp)/Q(a − ζp(a − 1)). For p = 3 this specialises to δ3(a) = N_Z[ω](a − ω(a − 1)), connecting the individual cubic finite difference — obtained by differencing the classical sum formula of Nicomachus of Gerasa (∼100 CE) — with the Eisenstein norm that appears in Euler's factorisation of a³ + b³.Starting from the historical identity S3(n) = Tₙ² where Tₙ = n(n + 1)/2, and applying the backward finite difference operator ∇f(n) := f(n) − f(n − 1) — formalised by Taylor (1715) and systematised by Boole (1860) — the Cubic Identity is derived: n³ = (n²/4)[(n + 1)² − (n − 1)²] = Tₙ² − Tₙ₋₁².This identity is extended to all p ≥ 1 via the Universal Faulhaber–Bernoulli Identity (UFBI): nᵖ = 1/(p+1) Σⱼ₌₀ᵖ C(p+1,j) Bⱼ⁺ δp+1−j(n), δm(n) := nᵐ − (n−1)ᵐ.The central contribution of this work is the Unified Chain Formula: ∇Tₙ² = δ3(a) = N_Z[ω](a − ω(a−1)) = Φ3(a, a−1) = N_Q(ζ3)/Q(a − ζ3(a−1)), which connects, in a single proved identity, five centuries of mathematics: Nicomachus (1st century), Boole (19th century), Euler/Eisenstein (18th century), and Gauss/cyclotomic theory (19th–20th centuries). This chain is not present as such in the existing literature; its originality lies in the explicit articulation of these connections, not in the individual equalities, each of which follows from classical results.Beyond the Unified Chain, the following new elements are introduced: (i) the Tower of Norms a³ = Σₖ₌₁ᵃ N(αk), making explicit how each perfect cube is a stack of hexagonal norms; (ii) the Cyclotomic Compatibility Index ICC(n, p), which quantifies the arithmetic obstruction to hᵖ = aᵖ + bᵖ having integer solutions; (iii) the Window Incompatibility Theorem, formalising why the hexagonal windows {a−1, a, a+1} and {b−1, b, b+1} can never merge into a single window {h−1, h, h+1} in Z[ω] for a, b ≥ 2; (iv) the Order Theorem for δm(n), providing a complete characterisation of prime divisibility of finite differences via multiplicative orders; (v) the Extreme Reduction Theorem (ERT), showing that the Order Filter eliminates every pair (a, b) with a ≥ 2 from the equation a³ + b³ = c³, reducing the problem to the case a = 1; (vi) the Fermatian Rigidity Index R(p), a quantitative measure of how far (aᵖ + bᵖ)^(1/p) is from an integer. All results are illustrated throughout by the single running example a = 6, b = 10, and the key number 91 = 7 × 13. Verified over 179,700 pairs with 50-digit precision: zero exceptions. This work does not claim to prove Fermat's Last Theorem, definitively established by Wiles [1].
For every prime p and every integer a, the backward finite difference δp(a) := aᵖ − (a − 1)ᵖ equals the cyclotomic binary form Φp(a, a − 1) and hence the norm N_Q(ζp)/Q(a − ζp(a − 1)). For p = 3 this specialises to δ3(a) = N_Z[ω](a − ω(a − 1)), connecting the individual cubic finite difference — obtained by differencing the classical sum formula of Nicomachus of Gerasa (∼100 CE) — with the Eisenstein norm that appears in Euler's factorisation of a³ + b³.Starting from the historical identity S3(n) = Tₙ² where Tₙ = n(n + 1)/2, and applying the backward finite difference operator ∇f(n) := f(n) − f(n − 1) — formalised by Taylor (1715) and systematised by Boole (1860) — the Cubic Identity is derived: n³ = (n²/4)[(n + 1)² − (n − 1)²] = Tₙ² − Tₙ₋₁².This identity is extended to all p ≥ 1 via the Universal Faulhaber–Bernoulli Identity (UFBI): nᵖ = 1/(p+1) Σⱼ₌₀ᵖ C(p+1,j) Bⱼ⁺ δp+1−j(n), δm(n) := nᵐ − (n−1)ᵐ.The central contribution of this work is the Unified Chain Formula: ∇Tₙ² = δ3(a) = N_Z[ω](a − ω(a−1)) = Φ3(a, a−1) = N_Q(ζ3)/Q(a − ζ3(a−1)), which connects, in a single proved identity, five centuries of mathematics: Nicomachus (1st century), Boole (19th century), Euler/Eisenstein (18th century), and Gauss/cyclotomic theory (19th–20th centuries). This chain is not present as such in the existing literature; its originality lies in the explicit articulation of these connections, not in the individual equalities, each of which follows from classical results.Beyond the Unified Chain, the following new elements are introduced: (i) the Tower of Norms a³ = Σₖ₌₁ᵃ N(αk), making explicit how each perfect cube is a stack of hexagonal norms; (ii) the Cyclotomic Compatibility Index ICC(n, p), which quantifies the arithmetic obstruction to hᵖ = aᵖ + bᵖ having integer solutions; (iii) the Window Incompatibility Theorem, formalising why the hexagonal windows {a−1, a, a+1} and {b−1, b, b+1} can never merge into a single window {h−1, h, h+1} in Z[ω] for a, b ≥ 2; (iv) the Order Theorem for δm(n), providing a complete characterisation of prime divisibility of finite differences via multiplicative orders; (v) the Extreme Reduction Theorem (ERT), showing that the Order Filter eliminates every pair (a, b) with a ≥ 2 from the equation a³ + b³ = c³, reducing the problem to the case a = 1; (vi) the Fermatian Rigidity Index R(p), a quantitative measure of how far (aᵖ + bᵖ)^(1/p) is from an integer. All results are illustrated throughout by the single running example a = 6, b = 10, and the key number 91 = 7 × 13. Verified over 179,700 pairs with 50-digit precision: zero exceptions. This work does not claim to prove Fermat's Last Theorem, definitively established by Wiles [1].
Posted: 06 March 2026
Predictors of Workplace AI Adoption in a Norwegian Sample
Sebastian Oltedal Thorp
,Lars Morten Rimol
,Martine Klock Fleten
,Simen Kristoffer Berg Hoel
Posted: 06 March 2026
Effects of Respiratory Vaccines in Older Adults with Cardiovascular Diseases: A Scoping Review
Fernando M. Runzer-Colmenares
,Nelson Luis Cahuapaza-Gutierrez
,Cielo Cinthya Calderon-Hernandez
,Mariam Miyanay Umeres-Bravo
Posted: 06 March 2026
On Feasibility of Quantum Computation and Quantum Communication
Guang-Liang Li
Posted: 06 March 2026
Recovery of Vocal Cord Function After Thyroid Surgery Often Takes Months with Loss of Signal During IONM as an Independent Predictor of Prolonged Recovery
Guglielmetti Laura
,Sina Schmidt
,Al-Hammoud Jasmin
,Senne Moritz
,Busch Mirjam
,Wagner Joachim
,Harsch Simone
,Andreas Zielke
,Smaxwil Constantin
Posted: 06 March 2026
Caregivers’ Baseline Mental Health Predicts Early Childhood Social Skills: Evidence from an Urban Area of Indonesia
Hilda Meriyandah
,Yuri Nurdiantami
,Smarika Shresta
,Maiko Shigeeda
,Tokie Anme
Posted: 06 March 2026
Telmisartan-Induced Alteration of Voltage-Gated Na+ Currents: Integrated Experimental and In Silico Approaches
Sheng-Nan Wu
,Rasa Liutkevičienė
,Vita Rovite
,Chung-Hung Tsai
,Sheng-Che Lin
Posted: 06 March 2026
Rhodamine B Dye–Coupled Hydrophobic Carbon Quantum Dots with Dual-Emission for White-Light Optoelectronic Applications
Alaa Y. Mahmoud
,Walaa Al-masri
Posted: 06 March 2026
Standardization of a Methodology for Disinfection Efficacy Against Microorganisms Isolated from a Pharmaceutical Industry Facility as a Contamination Control Strategy
Amanda Silva Costa
,Luciana Veloso da Costa
,Rebeca Vitória da Silva Lage de Miranda
,Talita Bernardo Valadão
,Stephen James Forsythe
,Marcelo Luiz Lima Brandão
Posted: 06 March 2026
Research on Ship Route Planning Based on an Improved Theta* Algorithm
Junwei Dong
,Ze Sun
,Peng Zhang
,Jiale Zhang
,Chen Chen
,Run Qian
Posted: 06 March 2026
Targeted Nanotechnology and AI-Driven Strategies to Penetrate Caseous Lesions and Overcome Immune Evasion in Tuberculosis: A Comprehensive Review
Amr Ahmed
Posted: 06 March 2026
Predictions of Land Use/Land Cover Changes, Drivers, and their Implications for Dense Forest Degradation in Kunar Province, Eastern Afghanistan
Bilal Jan Haji Muhammad
,Muhammad Jalal Mohabbat
,Lia Duarte
,Ana Cláudia Teodoro
Posted: 06 March 2026
Effects of Irrigation Depth on Graft Establishment, Nursery Productivity and Water Use Efficiency in Peach Nursery Production
Adelina Venig
Posted: 06 March 2026
Assessing the Accuracy and Clinical Potential of Virtual Reality-Based Visual Field Perimetry
Vivek Raghavan
Posted: 06 March 2026
The Influence of Professional Life Areas and Work Engagement on Burnout Syndrome among Psychologists
João Hipólito
,Tito Laneiro
,Samuel Antunes
,Yohana Souza
Posted: 06 March 2026
From Chebyshev to Primorials: Establishing the Riemann Hypothesis
Frank Vega
Posted: 06 March 2026
Quantitative-Qualitative Literature Review of the Contribution of Remote Sensing Air Pollution and Greenness to Stroke Subtype Occurrence and Risk
John T. Braggio
Posted: 06 March 2026
Enhancing BLS Methodologies for Projecting AI's Impact on Employment: A Data-Driven Framework for Measuring Labor Market Transformation
Satyadhar Joshi
Posted: 06 March 2026
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