Even Perfect Number is Triangular

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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.


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

The result follows from Closed Form for Triangular Numbers.


Proof 2

Follows from:

Even Perfect Number is Hexagonal
Hexagonal Number is Triangular Number