The Erdős–Straus Conjecture in Quaternionic Eulerian Calculus: The Area of Two, the Splitting of Unit Fractions, and Three Heights of the 1/𝑥 Curve

The Erdős–Straus conjecture asks that 4/𝑛 be a sum of three unit fractions for every 𝑛 ≥ 2. In Quaternionic Eulerian Calculus (QEC) the unit fractions are the heights of the curve 𝑦 = 1/𝑥 at the integers, each the derivative of ln 𝑥, and 4/𝑛 is the height of the triangle of base 𝑛 and area 2: [(4/𝑛) · 𝑛] ÷ 2 = 2. Equivalently, 4/𝑛 is the vertical leg of the tangent triangle of 1/𝑥 at 𝑥 = 𝑛/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/𝑚 = 1/(𝑚 + 1) + 1/(𝑚(𝑚 + 1)), read on the curve as the height at 𝑚 equalling the heights at 𝑚 + 1 and 𝑚(𝑚 + 1) together, proves the conjecture with explicit decompositions for every even 𝑛, every 𝑛 ≡ 3 (mod 4), every 𝑛 ≡ 2 (mod 3), and every multiple of these: every 𝑛 ≢ 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 𝑛 ≡ 1 (mod 12) is completed by the QEC principle of splitting. Every quantity is a pure number.