prime problem

**ribbie** · October 31st 2010, 12:29 AM

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

**Dinkydoe** · October 31st 2010, 01:50 AM

No, primefactorisation is unique. A statement like $p_ap_b = p_cp_d$ is a contradiction. If $n=m$ , then $n,m$ can be divided by the same primes. And so eventually $n,m$ get the same primefactorisation. You can't find a prime that divides $n$ , that does not divide $m$ . It's really trivial

If $p_1\cdots p_n= n$ . And suppose there's another element $p$ , which is not in the factorisation of $n$ , but does divide $n$ . Then $p|p_i$ for some prime. But then $p=p_i$ or $p=1$ .

**ribbie** · December 5th 2010, 03:53 AM

That's what I thought

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