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Is it proven for any integer $n$ that there are no perfect numbers of the form $kn+1$ or more simply it has been proven there are no perfect numbers congruent to $1$ $\pmod n$? For instance, $n$ $=$ $2$ would show that there are no perfect numbers of the form $2n+1$, or more simply there are no odd perfect numbers. Can someone come up with a proof for another number $n$ such that NO perfect numbers will exist of the form $kn+1$? ($n$ cannot be $3$ for instance since it is most likely that there are infinitely many even perfect numbers, numbers of the form $2^{n-1}(2^n-1)$, which all except $6$ be of the form $3n+1$. Thanks if such an integer is found and my case is proven for $n$.

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