3
$\begingroup$

(I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.)

Let $\sigma(M)$ be the sum of the divisors of the positive integer $M$. For example, $\sigma(6)=1+2+3+6=12$.

A number $N$ is called perfect if $\sigma(N)=2N$.

Euler proved that an odd perfect number $N$ must have the form $N=q^k n^2$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$. We call $q$ the Euler prime of $N$.

Here are my questions:

(1) Who is credited with making the conjecture that $q$ is the largest prime factor of $N$?

(2) Additionally, to whom should I attribute the conjecture that $k=1$? Should it be Descartes, Frenicle, or Sorli?

(Added September 28 2016 For (2): In an edit to this MO post, it is stated that (per Beasley), "Dris $\ldots$ refers to Descartes’ and Frenicle’s claim (that $k=1$) as Sorli’s conjecture; Dickson has documented Descartes’s conjecture as occurring in a letter to Marin Mersenne in 1638, with Frenicle’s subsequent observation occurring in 1657". As commented by Gerry, one would need to double-check Dickson's History of Number Theory to verify Beasley's statements.)

$\endgroup$
  • 2
    $\begingroup$ I don't know, but have you looked at Dickson's History of Number Theory? $\endgroup$ – Gerry Myerson Sep 11 '16 at 7:13
  • $\begingroup$ @GerryMyerson, will do. Thanks for pointing out that reference! =) $\endgroup$ – Jose Arnaldo Bebita-Dris Sep 11 '16 at 8:14
  • 1
    $\begingroup$ Doesn't your edit of 18 September 2013 on mathoverflow.net/questions/48203/… answer your (2) here? [and even if it doesn't, shouldn't you have at least linked to it?] $\endgroup$ – Gerry Myerson Sep 27 '16 at 23:34
  • 1
    $\begingroup$ My apologies for the oversight Gerry, doing so now. (By the way, I hope this question does not duplicate that other one...) $\endgroup$ – Jose Arnaldo Bebita-Dris Sep 27 '16 at 23:38

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.