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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.

Note by Alexander Israel Flores Gutiérrez
1 year, 4 months ago

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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 - 1 year, 4 months ago

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