On the Perfect Number Ratio Function and the Existence of an Odd Perfect Number: Part I: Elimination of Primes and Squares via Divisor Functions

Title: On the Perfect Number Ratio Function and the Existence of an Odd Perfect Number: Part I — Elimination of Primes and Squares via Divisor Functions Description: This is Part I of a planned four-part study of the odd perfect number (OPN) problem, built around the perfect number ratio function Γ(n) = S(n)/n, where S(n) is the sum of proper divisors of n. Part I lays the foundations: it proves that Γ(n) = 1 if and only if n is perfect, and establishes that Γ(n) is finite and non-vanishing for all n ≥ 2. It then partitions the odd integers K into three disjoint sets — odd primes P, odd perfect squares R, and odd composite non-squares D — and shows, via divisor-parity arguments, that no element of P or R can satisfy Γ(n) = 1. Combined with Euler's canonical form for a hypothetical odd perfect number, this reduces the search for an OPN entirely to the set D, with the additional restriction that any such number must be congruent to 1 (mod 4). This paper is foundational and self-contained; it does not claim novelty for the classical results it organizes (Euler's theorem, Sylvester's and Touchard's theorems), but reformulates them around Γ(n) as a framework for the subsequent parts of the series. Part II will introduce divisor-conjugate pairing on D, Part III will study the arithmetic of S(n) on D, and Part IV will introduce a Divisor Class Theorem addressing existence.