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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 $$k=2^n-1$$

- 1 year, 4 months ago