We know the even perfect number formula is $2^{p-1}(2^p − 1)$ and it is known that the multiplication of a even number and odd number is a even number. So why can't we say there are infinitely many perfect numbers since $2^p-1$ has to be a prime number in $2^{p-1}(2^p − 1)$ and it is known that there are infinitely many prime numbers?

Note: I'm not a native english speaker.


Because $2^{p}-1$ is not always a prime. When it is a prime, it is called a Mersenne prime.

For example, $11$ is prime, but $2^{11}-1 = 2047 = 23 \times 89$.

  • $\begingroup$ Ohh, I understood it now, thanks $\endgroup$ – Mehmet Sep 27 '14 at 16:26

If $2^p - 1$ is a prime (often called a Mersenne prime), then $2^{p-1}(2^p - 1)$ is a perfect number. While there are infinitely many primes, it is not known whether there are infinitely many Mersenne primes. There are many primes which cannot be written in the form $2^p - 1$, for example, the prime $5$.

  • $\begingroup$ I was doing a research and I saw the question "Are there infinitely many perfect numbers?" in the list of unsolved problems in mathematics(link: en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics) Should I assume this question was already solved? $\endgroup$ – Mehmet Sep 27 '14 at 16:22
  • $\begingroup$ It is still unsolved. I did not claim it is true or false, I just pointed out that while there are infinitely many primes, it is unknown whether there are infinitely many primes of the form $2^p - 1$. $\endgroup$ – Michael Albanese Sep 27 '14 at 16:28
  • $\begingroup$ Thank you for your help :) $\endgroup$ – Mehmet Sep 27 '14 at 17:17

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