# Show that, there do not exist two consecutive perfect numbers.

Show that, there do not exist two consecutive perfect numbers
I know that it is unknown whether there exist any odd perfect number(s) or not.
Also, I know that all the even perfect numbers are determined by Euler's theorem.
Which states that:

If $$N$$ is an even perfect number, then N can be written in the form $$N = 2^{n−1} (2^n − 1)$$, where $$2^n − 1$$ is prime

But, still, I am unable to crack the above-mentioned problem asked in some olympiad, I guess.

• Wikipedia lists many necessary conditions for an odd perfect number. Maybe, this helps to rule out numbers of the desired form. Jan 25, 2022 at 11:34

Suppose that $$E$$ and $$P$$ are consecutive perfect numbers with $$E$$ even. Since $$5$$ and $$7$$ are not perfect, $$E\ne 6$$. Then from Euclid's formula, $$E$$ is $$4$$ (mod $$12$$).

First suppose that $$P>E$$ and then $$P$$ is $$2$$ (mod $$3$$). Let $$x$$ be any divisor of $$P$$ and consider $$y=\frac{P}{x}$$. Then $$xy$$ is $$2$$ (mod $$3$$) and so $$x+y$$ is $$0$$ (mod $$3$$). Thus the divisors of $$P$$ can be paired so the sum of each pair is divisible by $$3$$. Hence $$\sigma (P)$$ is divisible by $$3$$ and so $$P$$ is not perfect.

Now suppose $$P and then $$P$$ is $$3$$ (mod $$4$$). The same argument proves that $$\sigma (P)$$ is divisible by $$4$$ and so, again, $$P$$ is not perfect.

Your question/problem is covered exactly in the following M. Sc. thesis, completed in 2008, at De La Salle University - Manila.

You may refer to pages 53 to 55.

Basically, the following facts about perfect numbers are used:

• A number $$M$$ (odd or even) satisfying $$M \equiv 2 \pmod 3$$ cannot be perfect.
• An odd perfect number $$N$$ must satisfy $$N \equiv 1 \pmod 4$$.

The proof proceeds by first showing that if $$O$$ is odd and perfect, then $$O - 1$$ and $$O + 1$$ cannot be a(n) (even) perfect number. Next, supposing that $$E$$ is even and perfect, we show that $$E - 1$$ and $$E + 1$$ cannot be a(n) (odd) perfect number.

This completes the proof.

• Care to explain the quick downvote? Jan 25, 2022 at 11:38
• Surely the second fact ($E-1$ and $E+1$ not perfect) is implied by the first ($O-1$ and $O+1$ not perfect)? Jan 25, 2022 at 11:44
• No, @TonyK, the implication does not follow. I presented the proof in that way so as to ensure that an even perfect number and an odd perfect number cannot differ by $1$. Jan 25, 2022 at 11:48
• @TonyK: You may refer to the papers by Luca, and Holdener, et. al for more details. The part of my thesis that I quoted in this answer is basically an exposition of their results on this topic. Jan 25, 2022 at 11:50
• I don't need to refer to any papers to see that the first of those facts implies the second. Jan 25, 2022 at 12:12