This is a question that interests me and I don't know how to find the answer: Do numbers exist that are simultaneously the product of two different pairs of primes? Can you give examples or proof?
I mean, can Pa * Pb = Pc * Pd?
No, primefactorisation is unique. A statement like is a contradiction. If , then can be divided by the same primes. And so eventually get the same primefactorisation. You can't find a prime that divides , that does not divide . It's really trivial
If . And suppose there's another element , which is not in the factorisation of , but does divide . Then for some prime. But then or .