For semi-prime c, is there an equation or system of equations that will find its factors?
In other words, if a prime number a time a prime number b equals semi-prime c, where c isn't one and neither a nor b are equal to one, how can I find a or b in terms of c?
In math notation, ab = c {a && b ∈ prime: c ∈ semi-prime}.
I attempted to solve it by making the observation that, since ab = c, b = c/a. This meant that we could do a substitution and find that (a^2)/c = c. The issue with this is it follows that a^2 = c^2, and therefore a = ±c and b = ±1, which is not valid as both a and b must be prime numbers.
Then I tried some other almost-valid proofs (meaning proofs that make unwarrented leaps in logic that are only sometimes accurate), that while interesting, don't add anything here because they only work if c is also a perfect square, and prove that a = b so a^2 = c which means that a = b = sqrt(c), which is only valid some of the time.
Finally, I tried doing some "dancing with substitution" stuff that just kept going in circles.
Is this problem just impossible, or is there a solution?