(I have asked a similar question in MSE around a week ago, but did not receive any responses. I have therefore cross-posted it to this site, hoping to get some answers.)

An odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime, $q \equiv k \equiv 1 \pmod 4$, and $\gcd(q, n) = 1$.

Here is my question:

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ squarefree?

I would certainly appreciate it if someone could point me to existing papers in the literature where this particular question is addressed.


No, if an odd perfect number exists, then $n$ must contain a square factor. This is a 1937 result of Steuerwald:

R. Steuerwald, "Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl," S.-B. Math.-Nat. Abt. Bayer. Akad. Wiss., 1937, pp. 68-73.

A very nice recent paper on this theme is by Fletcher, Nielsen, and Ochem (Math. Comp.); the preprint version is freely available on Nielsen's website: https://math.byu.edu/~pace/OPNSieves_web.pdf

[Edited as per Lucia's suggestion.]

  • $\begingroup$ It is absurd that someone chose to vote down this answer! (It took me a little effort to parse your answer -- maybe it would be clearer if the first line said something like: ``No: if an odd perfect number exists, then $n$ must necessarily contain a square factor.") $\endgroup$ – Lucia Dec 18 '15 at 22:33
  • $\begingroup$ Yes, that's clearer. I've made the edit.Thanks! $\endgroup$ – so-called friend Don Dec 19 '15 at 4:32
  • $\begingroup$ @so-calledfriendDon, mind if you check out this related MSE question, where I ask if $n$ is a square? Let me know if you need me to ask a separate MO question. $\endgroup$ – Arnie Bebita-Dris Dec 24 '15 at 14:10
  • $\begingroup$ @so-calledfriendDon, here is the new MO question. $\endgroup$ – Arnie Bebita-Dris Dec 27 '15 at 13:08

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.