Biconditional Inequality Links Euler Prime and Cofactor in Odd Perfect Numbers — E8 Intelligence Research

FINDING: Odd perfect numbers (OPNs), if they exist, must be of the form \(N = q^α ∏ᵢ₌₁ᵏ p_i2e_i\) with \(q ≡ α ≡ 1 {4}\) (Euler's theorem); recent work (arXiv:1309.0906v19) proves a biconditional involving the Euler prime \(q\) and the cofactor \(n^2\), specifically that \(q^k < n\) holds if and only if a certain divisor-sum inequality is satisfied, and this biconditional holds unconditionally. MATH: - Euler form: \(N = q^α ∏ᵢ₌₁ᵏ p_i2e_i\), \(q ≡ 1 {4}\), \(α ≡ 1 {4}\), \(p_i\) odd primes, \(e_i ≥ 1\). - Perfect number condition: \(σ(N) = 2N\), where \(σ\) is the sum-of-divisors function. - Key inequality from the paper: \(σ(q^k)/σ(n^2) ≤ 2\) and the biconditional: \(q^k < n σ(q^k) < σ(n^2)\) (under Dris's conjecture, later shown unconditional). - Known lower bound: \(N > 10¹⁵⁰⁰\) (Ochem & Rao), and \(q^α > 10⁶²\) (recent refinements). - No new Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com