Sierpiński's 5/𝑛 Conjecture in Quaternionic Eulerian Calculus: The Splitting of Unit Fractions on the 1/𝑥 Curve

Sierpiński's conjecture asks that 5/𝑛 be a sum of three unit fractions for every 𝑛 ≥ 2. It is the next member, after the Erdős–Straus conjecture for 4/𝑛, of the family of statements about 𝑘/𝑛. In Quaternionic Eulerian Calculus (QEC) the unit fractions are the heights of 𝑦 = 1/𝑥 at the integers, the derivatives of ln 𝑥, and 5/𝑛 is the vertical leg of the tangent triangle of constant area 2 at the point 𝑥 = 2𝑛/5. Three rules settle the conjecture with explicit decompositions for all but one class of 𝑛: every unit fraction is composed of two others, 1/𝑚 = 1/(𝑚 + 1) + 1/(𝑚(𝑚 + 1)); 2/𝑀 is two unit fractions for every 𝑀; and 3/𝑀 is two unit fractions whenever 𝑀 ≢ 1 (mod 3). With them 5/𝑛 is split for every even 𝑛, every multiple of 5, every 𝑛 ≡ 3 or 4 (mod 5), every odd 𝑛 ≡ 2 (mod 5), and every multiple of these: that is, for every 𝑛 having a prime factor not ending in the digit 1. The class that remains, 𝑛 whose prime factors all end in 1 (11, 31, 41, 61, 71, 101, 121, …), is shown to escape the single-step constructions, its first members are decomposed individually, and the class is completed by the QEC principle of splitting. All quantities are pure numbers.