even and odd perfect numbers existence

Thank for my previous post. Also, thank you so much for this site (m.s.e)

1) If odd perfect numbers there, those numbers can be expressible $12k + 1$ or $324k + 81$ or $468k + 117$. If yes, please discuss, how far I am correct.

2) If $K$ = $(4^n - 2^n)$/2 is perfect, when $k = 1^3 + 3^3 + ...$

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You might want to add how you came up with these results ... – Hagen von Eitzen Feb 3 '13 at 8:56
@HagenvonEitzen! sinece even perfect numbers cannot wirtien in 12k + 1 and so on,.. – Jiha Feb 3 '13 at 9:19
Well, even perfect numbers surely cannot be written as $12k+1$. What was your reasoning about odd perfect numbers? – Hagen von Eitzen Feb 3 '13 at 9:24

The formula $$\sum_{k=1}^nk^3=\frac{n^2(n+1)^2}{4}$$ is well-known and leads to $$1^3+3^3+\ldots+(2m-1)^3=\sum_{k=1}^{2m}k^3-\sum_{k=1}^{m}(2k)^3\\=\frac{(2m)^2(2m+1)^2}{4}-8\cdot\frac{m^2(m+1)^2}{4}\\=m^2(2m^2-1).$$ Since it is well-known that even perfect numbers $N$ are of the form $N=2^{p-1}(2^p-1)$ with $p=2n+1$ an odd prime and $2^p-1$ a Mersenne prime, letting $m=2^n$, you find that indeed $$N=\frac{4^n-2^n}{2}=m^2(2m^2-1)=1^3+3^3+\ldots+(2m-1)^3.$$