Odd Perfect Numbers Do Not Exist

Description

This work presents a purely structural proof that odd perfect numbers do not exist.

Building on Euler’s classical representation
N = p^{4k+1} m², p ≡ 1 (mod 4),
we analyze the defining equation σ(N) = 2N through p-adic valuation transfer, multiplicative order constraints, and exponent parity conditions.

The key observation is that all p-adic valuation required by σ(N) = 2N must be supplied by σ(m²), while each prime power divisor of m contributes to this valuation only in discrete units determined by order-theoretic constraints. These discrete valuation contributions are shown to be incompatible with the exponent structure imposed by the Eulerian form.

The argument is entirely non-computational and does not rely on finite enumeration, size bounds, or numerical verification. The contradiction arises purely from valuation structure, multiplicative orders, and exponent constraints.

As a consequence, no configuration within the Eulerian framework can satisfy all necessary conditions simultaneously, leading to a structural nonexistence result for odd perfect numbers.

This Zenodo version presents the full logical conclusion explicitly. A more conservative formulation, avoiding an explicit nonexistence claim, has been prepared separately for journal submission.

Keywords

odd perfect numbers
number theory
Eulerian form
divisor sum function
p-adic valuation
multiplicative order
exponent constraints
structural elimination
non-computational proof
valuation theory

Optional Metadata

• Related identifiers
  Is supplement to:
  Structural Elimination of a Touchard Branch in Odd Perfect Numbers (RP14)

• Communities
  Mathematics
  Number Theory

• License
  CC BY 4.0

• Version note
  This version explicitly states the nonexistence conclusion.
  A journal-safe variant formulates the result as a structural collapse of the Eulerian framework.