Prove that a Perfect Number is Triangular.

Let $N$ be a Perfect Number (a positive integer that is equal to the sum of its proper positive divisors), like $6$ , or $28$ . It seems that all known perfect numbers are the sum of a series of consecutive positive integers starting with $1$ , that is, it seems that any perfect number is a triangular number. For example:
$6=1+2+3$ ,
$28=1+2+3+4+5+6+7$ .

Prove that any even perfect number is triangular.

Note: It is not known whether there are any odd perfect numbers.

#NumberTheory

Easy Math Editor

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`2^{34}`	$2^{34}$
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Comments

Any even perfect number can be expressed as $2^{n-1}(2^n-1)$ for some natural number $n$ .Now it's quite easily seen that $2^{n-1}(2^n-1)=\dfrac{2^n(2^n-1)}{2}$ which fits the expression $\dfrac{k(k+1)}{2}$ (this is the expression for all triangular numbers) when $k=2^n-1$

Abdur Rehman Zahid - 4 years, 10 months 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}`	$\boxed{123}$