The ErdΕsβStraus Conjecture in Quaternionic Eulerian Calculus: The Area of Two, the Splitting of Unit Fractions, and Three Heights of the 1/π₯ Curve
Year: 2026
Type: article
Abstract: The ErdΕsβStraus conjecture asks that 4/π be a sum of three unit fractions for every π β₯ 2. In Quaternionic Eulerian Calculus (QEC) the unit fractions are the heights of the curve π¦ = 1/π₯ at the integers, each the derivative of ln π₯, and 4/π is the height of the triangle of base π and area 2: [(4/π) Β· π] Γ· 2 = 2. Equivalently, 4/π is the vertical leg of the tangent triangle of 1/π₯ at π₯ = π/2, the triangle of constant area 2. The conjecture therefore asks that one leg of an area-two triangle be composed of three heights of the curve. A single rule, that every unit fraction is composed of two others, 1/π = 1/(π + 1) + 1/(π(π + 1)), read on the curve as the height at π equalling the heights at π + 1 and π(π + 1) together, proves the conjecture with explicit decompositions for every even π, every π β‘ 3 (mod 4), every π β‘ 2 (mod 3), and every multiple of these: every π β’ 1 (mod 12). Why that one class resists the single-step constructions is shown. The demonstrated results of Mordell, Vaughan and Salez are set beside the proof, and the class π β‘ 1 (mod 12) is completed by the QEC principle of splitting. Every quantity is a pure number.
Author: Willie Johnson Jr.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225797313
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