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(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is appropriate for this site.

I would like to request references to research done as to whether the Euler prime of an odd perfect number can also be its largest factor.

To be more specific, the Euler prime $q$ of an odd perfect number $N$ is the sole prime factor that occurs to an (odd) exponent $k \equiv 1 \pmod 4$. That is, we can write this odd perfect number in the form $N = {q^k}{n^2}$, where $\gcd(q, n) = 1$.

In an e-mail, it was communicated to me by Douglas Iannucci that his adviser, Peter Hagis Jr., considered this possibility.

I was wondering if anybody here knows of any partial results in this direction.

Thank you!

[Added Feb 8 2015] We do know that the Euler prime $q$ is not the smallest prime factor of an odd perfect number $N = {q^k}{n^2}$. To see why, it suffices to consider:

$$q + 1 = \sigma(q) \mid \sigma(q^k) \mid 2N.$$

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  • $\begingroup$ Perhaps give a good reason for your very quick downvote? =) $\endgroup$ – Jose Arnaldo Bebita-Dris Feb 11 '15 at 19:05
  • $\begingroup$ All votes are anonymous for a reason. Asking for a reason is ok, but I just wanted you to know that this has been discussed and people think anonymity is good. See this meta discussion for details: meta.mathoverflow.net/q/828/55893 (And no, I didn't vote in either direction.) Perhaps the downvoter hints that this question is not appropriate for MO. $\endgroup$ – Joonas Ilmavirta Feb 11 '15 at 19:21
  • $\begingroup$ Maybe. But based from the length of time it took the voter to downvote, I am guessing he/she did not bother to even check if results for which I am requesting a reference are well-known/easily searchable in Google/readily available. Indeed, a quick survey of search results in Google for "largest prime factor" "odd perfect number" "Euler prime" would return my own papers @LOL@. $\endgroup$ – Jose Arnaldo Bebita-Dris Feb 11 '15 at 19:26
  • $\begingroup$ I also found Samuel Dittmer's Spoof Odd Perfect Numbers. =) $\endgroup$ – Jose Arnaldo Bebita-Dris Mar 5 '15 at 2:35
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This paper by Bill Banks et al. studies spoof OPN's similar to Descartes' spoof, and of course in Descartes' spoof the "quasi" Euler prime is the biggest prime.

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    $\begingroup$ Thanks for the link. Do you or does Bill have an idea about odd multiperfects? I'm looking for literature which may touch on the idea in mathoverflow.net/questions/134826/… , and anything which has a similar smell to it. $\endgroup$ – The Masked Avenger Feb 12 '15 at 17:28
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    $\begingroup$ @TheMaskedAvenger: In my recent paper "Odd perfect numbers, Diophantine equations, and upper bounds" (which you can find on my website) I prove an upper bound on odd multi-perfect numbers in terms of the number of distinct prime factors. You may find that useful. $\endgroup$ – Pace Nielsen Feb 12 '15 at 18:04
  • $\begingroup$ I also found Samuel Dittmer's Spoof Odd Perfect Numbers. =) $\endgroup$ – Jose Arnaldo Bebita-Dris Mar 5 '15 at 2:35

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