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.

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TopNewestAny 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 · 10 months, 1 week ago

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