I have found a proof that there does not exist any odd perfect number, Am I Correct?

Question

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I really cannot imagine I did it, My Intuition to send this here is just please God there must be a mistake in here, but if magically there is isn't, I will really say that I will die out of happiness.

Answer 1

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Note that, following your logic, if $$2 = \frac{p^{k+1} - 1}{p - 1},$$ then necessarily $p$ is the special prime (satisfying $p \equiv k \equiv 1 \pmod 4$), which means that $\sigma(p^k) = 2$. We do know that $\sigma(p^k) \equiv 1 + \ldots + 1 \equiv k + 1 \pmod 4$ (since $p \equiv 1 \pmod 4$), and then $\sigma(p^k) \equiv 2 \pmod 4$ (since $k \equiv 1 \pmod 4$). (Looks good so far.) However, notice that since $p$ is the special prime satisfying $p \equiv 1 \pmod 4$ and since $k \equiv 1 \pmod 4$ implies that $k \geq 1$, then $$\sigma(p^k) \geq p^k + 1 \geq 6.$$ This lower bound for $\sigma(p^k)$ clearly contradicts your result $\sigma(p^k)=2$.