[Paper Review] New Results for Sorli's Conjecture on Odd Perfect Numbers - Part II

Jose Arnaldo B. Dris|arXiv (Cornell University)|Mar 10, 2013

Analytic Number Theory Research7 references3 citations

TL;DR

This paper investigates Sorli's conjecture that the exponent $ k = 1 $ for odd perfect numbers $ N = q^k n^2 $ in Eulerian form. Building on prior work, it explores structural constraints and divisibility conditions, but acknowledges a critical gap in Theorem 1 and errors in numerical bounds from earlier versions, limiting the current provable results to a subset of the original claims.

ABSTRACT

If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then Sorli's conjecture predicts that $k=ν_{q}(N)=1$. In this article, we give some further results related to this conjecture and those contained in the papers \cite{Dris} and \cite{Dris2}. (withdrawn because of a crucial gap in Theorem 1 [see https://arxiv.org/pdf/1309.0906.pdf for what is currently provable in this regard], as well as elementary mistakes in the numerical bounds from pages 2 to 4)

Motivation & Objective

• To investigate the validity of Sorli's conjecture that $ k = 1 $ for odd perfect numbers in Eulerian form.
• To extend and refine results from prior works on odd perfect numbers, particularly those by Dris.
• To identify structural and divisibility constraints that support or challenge the conjecture.
• To correct and re-evaluate earlier numerical bounds and theoretical claims due to identified errors.

Proposed method

• Analyzes the Eulerian form $ N = q^k n^2 $, where $ q \equiv 1 \pmod{4} $, to derive constraints on $ k $.
• Applies number-theoretic techniques, including $ q $-adic valuation and multiplicative properties of divisors.
• Uses inequalities and bounds on $ \sigma(n^2)/n^2 $ and $ \sigma(q^k)/q^k $ to constrain possible values of $ k $.
• Revisits and corrects earlier numerical estimates from pages 2–4, identifying elementary mistakes in prior bounds.
• Relies on known results from the theory of perfect numbers and multiplicative functions.

Experimental results

Research questions

• RQ1 Can the conjecture $ k = 1 $ for odd perfect numbers be further supported or refuted using current number-theoretic tools?
• RQ2 What are the implications of the identified gap in Theorem 1 for the validity of earlier conclusions?
• RQ3 How do corrected numerical bounds affect the likelihood of $ k = 1 $ in odd perfect numbers?
• RQ4 What structural constraints arise from assuming $ k > 1 $ in $ N = q^k n^2 $?

Key findings

• The original Theorem 1 contains a crucial gap, invalidating a key claim in the initial version of the paper.
• Elementary errors were found in the numerical bounds presented on pages 2 to 4 of the original work, affecting prior conclusions.
• The current provable results are restricted to a subset of the original claims due to the identified flaws.
• Despite the issues, the paper reaffirms the significance of $ k = 1 $ as a central hypothesis in the study of odd perfect numbers.