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?
TIA
No, primefactorisation is unique. A statement like $\displaystyle p_ap_b = p_cp_d$ is a contradiction. If $\displaystyle n=m$, then $\displaystyle n,m$ can be divided by the same primes. And so eventually $\displaystyle n,m$ get the same primefactorisation. You can't find a prime that divides $\displaystyle n$, that does not divide $\displaystyle m$. It's really trivial
If $\displaystyle p_1\cdots p_n= n$. And suppose there's another element $\displaystyle p$, which is not in the factorisation of $\displaystyle n$, but does divide $\displaystyle n$. Then $\displaystyle p|p_i$ for some prime. But then $\displaystyle p=p_i$ or $\displaystyle p=1$.