(Note: This has been cross-posted to MO.)

Good day!

I would like to request for 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 advisor, 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:

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

  • $\begingroup$ Thank you for the first up-vote! =) $\endgroup$ – Arnie Bebita-Dris Feb 7 '15 at 22:42
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    $\begingroup$ @barakmanos, since we still do not know whether or not an odd perfect number exists, the logical way to prove that there is none is to show a contradiction from assuming some (myriad of) conditions. Now if it so happens that there does exist an odd perfect number (currently known to be at least ${10}^{2000}$), then research on the largest prime factor might shed light on where to look for. That being said, thank you for your comment... =) $\endgroup$ – Arnie Bebita-Dris Feb 7 '15 at 23:09
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    $\begingroup$ @barakmanos, if you're still skeptical, I invite you to peruse the papers on odd perfect numbers by Hagis and Iannucci. $\endgroup$ – Arnie Bebita-Dris Feb 7 '15 at 23:11
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    $\begingroup$ Anyways, +1 for the interesting post... $\endgroup$ – barak manos Feb 7 '15 at 23:19
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    $\begingroup$ Actually, knowledge on odd perfect numbers can potentially yield information about even perfect numbers (and vice-versa). To see the connection, I humbly invite you to check out Euclid-Euler Heuristics for (Odd) Perfect Numbers. $\endgroup$ – Arnie Bebita-Dris Feb 7 '15 at 23:20

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