A short proof for the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

TL;DR

This paper provides a concise proof confirming the Descartes-Frenicle-Sorli conjecture, which states that in the Eulerian form of an odd perfect number, the exponent of the special prime is always one.

Contribution

The paper offers a new, short proof of the longstanding conjecture regarding the prime exponent in the Eulerian form of odd perfect numbers.

Findings

Confirmed the conjecture that the prime exponent is one in the Eulerian form of odd perfect numbers.

Provided a concise proof simplifying previous approaches.

Strengthened understanding of the structure of odd perfect numbers.

Abstract

If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then the Descartes-Frenicle-Sorli conjecture predicts that $k=\nu_{q}(N)=1$. In this article, we give a short proof for this conjecture.