SierpiΕski's 5/π Conjecture in Quaternionic Eulerian Calculus: The Splitting of Unit Fractions on the 1/π₯ Curve
Year: 2026
Type: article
Abstract: SierpiΕski's conjecture asks that 5/π be a sum of three unit fractions for every π β₯ 2. It is the next member, after the ErdΕsβStraus conjecture for 4/π, of the family of statements about π/π. In Quaternionic Eulerian Calculus (QEC) the unit fractions are the heights of π¦ = 1/π₯ at the integers, the derivatives of ln π₯, and 5/π is the vertical leg of the tangent triangle of constant area 2 at the point π₯ = 2π/5. Three rules settle the conjecture with explicit decompositions for all but one class of π: every unit fraction is composed of two others, 1/π = 1/(π + 1) + 1/(π(π + 1)); 2/π is two unit fractions for every π; and 3/π is two unit fractions whenever π β’ 1 (mod 3). With them 5/π is split for every even π, every multiple of 5, every π β‘ 3 or 4 (mod 5), every odd π β‘ 2 (mod 5), and every multiple of these: that is, for every π having a prime factor not ending in the digit 1. The class that remains, π whose prime factors all end in 1 (11, 31, 41, 61, 71, 101, 121, β¦), 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.
Author: Willie Johnson Jr.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225806121
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