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Gowers argues that even without considering generalizations of the reals, it is not "obvious". He argues that if the Theorem were "obvious", we should quickly be able to say that 23 x 1759 != 53 x 769, without multiplying them out.

Perhaps I'm dim but that seems like a ludicrous argument. The inequality is obvious, if you know that all the numbers are prime, and it's not if you don't. Ergo, what's non-obvious is that the big numbers are prime, not that fundamental theorem is true.

Shouldn't the standard of proof should be something like whether it's obvious that:

    23 * 7 * 11  =/=  17 * 7 * 13
? In this case the inequality is obvious, to the extent that you can convince yourself of it without multiplying anything. Doesn't this imply that therefore the theorem is obvious?

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".

But that's a weak argument. By that token, checking whether two numbers add up to a third or whether two numbers are equal or even whether two numbers have the same number of digits aren't obvious, if you make the numbers large enough.

"Can be answered quickly" is not a good indicator of "is obvious".

Well, yeah. Having the same sum, or having the same number of digits aren't really obvious properties of numbers.

Similarly, both being prime and having the same product aren't obvious properties either. Making the fundamental theorem of arithmetic non-obvious (or at least individual cases of it, but if the general case were obvious then the individual cases should also be obvious).

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