# Even Perfect Number is Triangular

## Theorem

All perfect numbers which are even are triangular.

## Proof 1

Let $a$ be an even perfect number.

From the Theorem of Even Perfect Numbers, $a$ is in the form $2^{p - 1} \left({2^p - 1}\right)$ where $2^p - 1$ is prime.

Thus:

 $\ds a$ $=$ $\ds \left({2^p - 1}\right) 2^{p - 1}$ $\ds$ $=$ $\ds \left({2^p - 1}\right) \frac {2^p} 2$ $\ds$ $=$ $\ds \frac {n \left({n + 1}\right)} 2$ where $n = 2^p - 1$

The result follows from Closed Form for Triangular Numbers.

$\blacksquare$

## Proof 2

Follows from:

Even Perfect Number is Hexagonal
Hexagonal Number is Triangular Number

$\blacksquare$