Indeed a general form!!

Solving the last problem made you observe that the first 3 perfect numbers are of the $2^{k-1}(2^{k} - 1)$ .

Actually it was proved by Euclid that :

If ( $2^{k}-1)$ is a prime the $n = 2^{k-1}(2^{k}-1)$ is perfect and every perfect number of this form.

Try to prove it in the comments box :)

You can also see that this implies all even perfect numbers are triangular numbers.

Also no odd perfect number has been found so far.

#NumberTheory #CosinesGroup

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Comments

Let $2^k-1=M_p$ [short for Mersenne Prime].

The positive divisors of $n=2^{k-1}(2^k-1)$ are $1$ , $2$ , $2^2$ , $2^3$ $\cdots$ $2^{k-1}$ and ${M_p}, 2M_p, 2^2 M_p, 2^3 M_p, \cdots 2^{k-1} M_p$ .

Let's add them all together.

We get $\sigma(n)=2^k-1 + M_p \times (2^k-1)=(2^k-1)(1+M_p)=(2^k-1)\times 2^k=2\times2^{k-1}(2^k-1)=2n$ .

So, $n$ is a perfect number by definition.

Mursalin Habib - 7 years, 1 month ago

When I first found out about perfect numbers (I was 9), I strived to find a pattern between them. After 2 hours only I figured it out but then checked the Internet. Turns out it was discovered by Euclid over 2000 years ago. I still felt proud.

Sharky Kesa - 7 years, 1 month ago

It was proven by Euler that every even perfect number must have this form.

Bogdan Simeonov - 7 years, 1 month ago

I think you're right ..but the general form was given by Euclid right??

Eddie The Head - 7 years, 1 month ago

Yes, Euclid found out that $\dfrac{p(p+1)}{2}$ was a perfect number for certain prime numbers $p$ . He concluded that those prime numbers had to be of the form $2^k-1$ . These primes are now called Mersenne Primes.

Euler later on proved that all even perfect numbers must have this form.

Mursalin Habib - 7 years, 1 month ago

@Mursalin Habib – All even prime numbers?

Sharky Kesa - 7 years, 1 month ago

@Sharky Kesa – Oops! Typo!

Mursalin Habib - 7 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$