Even Perfect Numbers are Triangular

Theorem

All perfect numbers which are even are triangular.

Proof

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:

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

The result follows from Closed Form for Triangular Numbers.

$\blacksquare$