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The following (interesting) assertion appears in the OeisWiki page on multiply-perfect numbers:

---------- Forwarded message ----------
From: Georgi Guninski <guninski@guninski.com>
To: Sequence Fanatics Discussion list <seqfan@list.seqfan.eu>
Cc: 
Date: Mon, 16 Jul 2012 13:14:33 +0300
Subject: [seqfan] Re: Reference that "A027687 4-perfect numbers" is finite
Thank you.

Asked because an odd perfect number and infinitely mersenne primes implies
4-perfect numbers are infinite (and a lot of other 2k-perfect numbers) -
take the product of the OPN and coprime to it EPN.

On the other hand 4-perfect being finite and infinitely mersenne primes
implies no OPN.

What is the reason to believe all 4-perfect are discovered (even if they
are finite)?

(This post is taken from the following SeqFan thread.)

Honestly, I cannot seem to wrap my head around the first assertion (and therefore, also the second).

Why is it that the existence of an odd perfect number and infinitely many Mersenne primes implies that there are infinitely many $4$-perfect numbers (and a lot of other $2k$-perfect numbers)?

It says "take the product of the OPN and coprime to it EPN".

However, an OPN and an EPN may not always be coprime, as a Mersenne prime (for example, $3$) may divide an OPN. (See this MSE question.)

Note: OPN = Odd Perfect Number, EPN = Even Perfect Number

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    $\begingroup$ What is "OPN" and "EPN" ? $\endgroup$ – Peter Dec 24 '18 at 11:55
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    $\begingroup$ @Peter: OPN = Odd Perfect Number, EPN = Even Perfect Number $\endgroup$ – Jose Arnaldo Bebita-Dris Dec 24 '18 at 12:08
  • $\begingroup$ Do you only want a proof that the existence of infinite many Mersenne primes and some odd perfect number implies that infinite many $4$-perfect numbers exist ? Or do you want to know something else ? $\endgroup$ – Peter Dec 24 '18 at 12:27
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    $\begingroup$ I think a proof for the implication in your last comment would suffice, @Peter. $\endgroup$ – Jose Arnaldo Bebita-Dris Dec 24 '18 at 12:32
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Suppose, an odd perfect number $k$ exists and infinite many Mersenne primes exist.

Let $m$ be a positive integer , such that $2^{m+1}-1$ is a Mersenne prime greater than $k$. With our assumption, infinite many such $m$ exist.

Now, consider $$N=2^m\cdot (2^{m+1}-1)\cdot k$$

We get $$\sigma(N)=(2^{m+1}-1)\cdot 2^{m+1}\cdot \sigma(k)=(2^{m+1}-1)\cdot 2^{m+2}\cdot k=4N$$ implying that infinite many $4$-perfect numbers exist.

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    $\begingroup$ Hence, if we assume that only finite many $4$-perfect numbers exist, but infinite many Mersenne-primes, we can conclude that no odd perfect number exists. $\endgroup$ – Peter Dec 24 '18 at 12:40
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    $\begingroup$ Indeed! Thanks, @Peter. =) $\endgroup$ – Jose Arnaldo Bebita-Dris Dec 24 '18 at 12:44

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