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 \(N = q^\alpha \prod_{i=1}^{k} p_i^{2e_i}\) with \(q \equiv \alpha \equiv 1 \pmod{4}\) (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: \(N = q^\alpha \prod_{i=1}^{k} p_i^{2e_i}\), \(q \equiv 1 \pmod{4}\), \(\alpha \equiv 1 \pmod{4}\), \(p_i\) odd primes, \(e_i \geq 1\). - Perfect number condition: \(\sigma(N) = 2N\), where \(\sigma\) is the sum-of-divisors function. - Key inequality from the paper: \(\sigma(q^k)/\sigma(n^2) \leq 2\) and the biconditional: \(q^k < n \iff \sigma(q^k) < \sigma(n^2)\) (under Dris's conjecture, later shown unconditional). - Known lower bound: \(N > 10^{1500}\) (Ochem & Rao), and \(q^\alpha > 10^{62}\) (recent refinements). - No new Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com