Hi, I apologize if there is already an (obvious) answer to my question, but please bear with me for the moment as I find it hard to see a good way to answer this question:

In the same way that the Mersenne primes uniquely determine the even perfect numbers (i.e. in the "bijective" sense), do the Euler primes likewise uniquely determine the odd perfect numbers?

That is, while an Euler prime obviously maps to a single odd perfect number, it is not obvious to me that an odd perfect number could only map to a single Euler prime.

This is because:

(1) In the most general case, it is not known whether squares are friendly or solitary.

(2) It is not even known if the smallest possible Euler prime 5 does divide an odd perfect number.

(3) It is conceivable that two distinct numbers $P$ and $Q$ which satisfy a particular number-theoretic equation do not necessarily have to satisfy $\gcd(P, Q) = 1$.

I hope you guys can help me out on this one. Establishing this very last gap will provide a proof for the implication:

Assuming Sorli's Conjecture for OPNs is true, then there are no OPNs.