Odd perfect numbers do not exist. According to Euclid's formulae a perfect number is equal to $2^P(2^P-1$) where $2^P-1$ should be prime. Only $2$ is the even prime number and rest are odd. So $2^P-1$ is odd. $(2^P-1) +1= 2^P$

That is $2^P-1$ is odd , an odd number $+ 1$ is even . Therefore $2^P$ is even. Product of odd and even is always even. Since $2^P-1$ is odd and $2^P$ is even $2^P(2^P-1)$ is even. That is odd perfect numbers do not exist.


  1. Determining if any odd perfect numbers exist.
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    $\begingroup$ You are not correctly quoting what the Euclid-Euler Theorem says. It says that $2^p(2^p-1)$ is an even perfect number if $2^p-1$ is prime. That doesn't mean that's all of them, just the even ones. $\endgroup$ – Hayden Sep 4 '17 at 13:38
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    $\begingroup$ This is somewhat an insult to other people, especially mathematicians. If you thought people wouldn't think of something like that. I don't mean to insult you, for the future times before you decide you came up with something think of this fact. $\endgroup$ – JukesOnYou Sep 4 '17 at 13:40
  • $\begingroup$ @JukesOnYou I have to say, I don't like thinking in this way. I think logical thought, regardless of how impertinent, ought to be fostered. If people were concerned about insulting the great minds that have come before them, how would any long-time open problems ever be solved? $\endgroup$ – Theo Bendit Sep 4 '17 at 13:49
  • $\begingroup$ @TheoBendit By developing new theories that didn't exist by the time those great minds lived? $\endgroup$ – JukesOnYou Sep 4 '17 at 14:20
  • $\begingroup$ @JukesOnYou I guess that's one way, but it's not something that one can just jump into. In order to approach a problem properly, one typically studies the obvious approaches. If what seems like a proof presents itself, there's nothing wrong with presenting it, IMHO. I myself am working on a 70 year old open problem for my PhD, and I've had to learn to silence that voice that tells me my results aren't good enough, because "surely someone before me would have noticed this". $\endgroup$ – Theo Bendit Sep 4 '17 at 14:27

You're wrong. All that Euclid proved was that when $2^p-1$ is prime, then $2^{p-1}(2^p-1)$ is perfect. He never said that all perfect numbers can be obtained by this process.

In the XVIIIth century, Euler proved that every even perfect number can be obtained by that process. He never stated that this holds for odd perfect numbers too.

  • $\begingroup$ Later Euler proved that you do get all the even perfect numbers this way. Of course that leaves the existence of odd perfect numbers an open question. $\endgroup$ – Ethan Bolker Sep 4 '17 at 13:46
  • $\begingroup$ @EthanBolker I know that. I wrote what I wrote because the OP wrote about “Euclid's formulae”. $\endgroup$ – José Carlos Santos Sep 4 '17 at 13:49
  • $\begingroup$ I pretty much know you knew that. I just thought it should be part of the answer even though it's more than was asked for. $\endgroup$ – Ethan Bolker Sep 4 '17 at 13:51
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    $\begingroup$ To say that Euler proved nothing about odd perfect numbers is a bit misleading, is it not? For one, Euler was the one who proved that an odd perfect number $N$ must have the form $q^k n^2$, where $k$ was conjectured to be $1$ by Descartes, Frenicle, and more recently, by Sorli. We still get to use Euler's theorem on the form of an odd perfect number to this day. $\endgroup$ – Jose Arnaldo Bebita-Dris Sep 18 '17 at 1:06
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    $\begingroup$ @JoseArnaldoBebitaDris I thought that what I meant was clear from the context, but I edited that sentence. $\endgroup$ – José Carlos Santos Sep 18 '17 at 7:39

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