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It is only obvious if you take the theorem as a given (as mathematicians typically do). However, I bet if you are given a slightly different system, it would be difficult to say whether this is true: For some natural number N, do there exist n primes p1,..,pn such that p1..pn = N and no other m primes q1,..,qm where q1..qm = N ? As an example, Gowers gives the complex numbers, where it may intuitively appear to be true ... but it is not.

Gowers mentions that, to him, an obvious argument would be: 23 x 22 != 21 x 25, since 2 divides 23x22, but 2 does not divide 21x25.




> Gowers mentions that, to him, an obvious argument would be: 23 x 22 != 21 x 25, since 2 divides 23x22, but 2 does not divide 21x25.

I don't follow the distinction - isn't the same true of my example, just swapping "2" for "11" or "23"?

I mean, given Gowers' standing I'm prepared to assume his premise is true, and I didn't follow the last argument (by comparison to other kinds of rings) at all so I'm guessing that it's the "real" reason why the theorem isn't obvious. But the preceding arguments made no sense to me at all.


The difference is that divisibility by 2 can be tested almost instantly. (Only the least significant digits of each factor matter.) However, testing each factor for divisibility is often as tedious as the original multiplication!

I think we're all at a loss for a good definition of "obvious".




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