Cyclotomic Squareclass Exclusions, Diophantine Covers, and Sigma-graph Constraints for Odd Perfect Numbers
Year: 2026
Type: article
Abstract: This paper develops a conditional structural framework for odd perfect numbers N under the hypothesis that 3^2 exactly divides N, exactly two non‑special components have exponent greater than 2, and one of those components has base 97. The analysis is organised into three interconnected parts: a complete 3‑adic valuation ledger and a normalised sigma‑graph connectivity theorem; a smooth‑support partition bound and capacity‑priced tail mechanism that exclude seven residual role assignments; and detailed Diophantine reductions of the remaining exceptional cases. The first part establishes an exact two‑sided cut identity for the normalised weighted sigma graph of any odd perfect number, proving that the graph is strongly connected—every vertex and every edge lies on a directed cycle. This converts factor‑chain propagation into a global recycling problem and rules out componentwise rank arguments as a termination mechanism. A seed‑free defect‑core theorem shows that when v_3(N) is 0 or 2 and only two non‑special components have exponent exceeding 2, the core has at most six vertices; exact capacity tests and source‑receiver ledgers reduce the problem to finite‑dimensional valuation systems. The second part develops two upper‑bound mechanisms for ordinary Phi_3‑rows. A smooth‑support partition bound controls the infinite tail of S‑smooth exponent vectors, while a shared‑price capacity dual with a finite‑state dynamic program incorporates residual valuation capacities. Exact rational catalogues and finite enumerations then conditionally exclude the special‑13 roles h = 19, 127, both special‑61 roles h = 31, 331, the active high‑13 continuation, and the active high‑61 continuation. The surviving roles reduce to four classes: the special‑13, h = 7 role; the inactive high‑13 continuation; the inactive high‑61 continuation; and the saturated 13^2, 61^2 || N architecture. In the special‑13, h = 7 survivor, every survivor contains an ordinary prime q satisfying q^2 + q + 1 = 7^a * 13^b * 97^c with primitive exponent vector. This equation is covered by 27 Mordell curves (26 of which can contain an admissible point after the primitive‑row exclusion), by at most 48 Eisenstein unit‑orientation cases, and by seven Pell families of norm −3 yielding exactly fourteen positive unit orbits. Exact recurrence enumeration shows no admissible prime occurs for q ≤ 10^50000. An independent exponent‑11 Frey reduction gives 2660 parity‑refined lanes at fourteen residual levels; exact trace comparisons exclude all 91 rational‑coefficient newform orbits, while nonrational orbits of total degree 2852 remain. For the saturated 3^2, 13^2, 61^2, 97^e architecture with exactly one additional higher component h^f, the paper proves h < 47500, excludes every prime divisor of e+1 below 101, and shows v_61(sigma(97^e)) = 0. A telescoping cyclotomic factorisation establishes that Omega(e+1) is bounded by the number of exceptional squarefree roles. The special‑prime‑free squareclass reduces to sigma(97^e) = h * y^2 with e+1 prime; an exhaustive 358‑pair catalogue and quadratic‑nonresidue witnesses exclude every case. The surviving squareclasses therefore contain the special prime and have prime or semiprime index. All exclusions are conditional on the stated two‑higher‑component architecture and do not prove the nonexistence of odd perfect numbers. Several residual Diophantine problems remain open, including the complete solution of the fixed‑support equation in the h = 7 role, the inactive continuations, and the saturated squareclass cases with epsilon_p = 1.
Author: Fathi, Kevin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7141390466
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