This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd perfect number with special / Euler prime $q$ (satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$). (Specifically, via exhaustion of all possible cases, it appears plausible that $k>1$ must hold true.)

On the other hand, for the Descartes spoof (which is probably the closest we will ever get to an "actual odd perfect number") $$\mathscr{D} = {3^2}\cdot{7^2}\cdot{{11}^2}\cdot{{13}^2}\cdot{22021} = 198585576189$$ the quasi-Euler prime $22021$ (which is actually composite) has exponent $1$. Additionally, the non-Euler part ${3^2}\cdot{7^2}\cdot{{11}^2}\cdot{{13}^2}$ of $\mathscr{D}$ can be (directly) shown to be deficient-perfect, a condition which is known to be equivalent to the Descartes-Frenicle-Sorli conjecture.

So here is my question:

What other evidence or heuristic do you know of that supports (or otherwise does not support) the Descartes-Frenicle-Sorli conjecture for odd perfect numbers?

One piece of evidence/heuristic per answer only, please.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.