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I hope this isn't a stupid question...

It's well known that (in the presence of various other axioms), Euclid's Postulate 5 ('parallel axiom') is equivalent to the Pythagorean Theorem. That is, assume Pyth. Thm. is true without Postulate 5, and you get the 'parallel axiom' as a theorem.

My question: Are there well-known (or not-so-well-known) theorems/properties of the ring of integers which are equivalent to the Fundamental Theorem of Arithmetic in this way? That is, things which are not just consequences of it, but imply it.

I have had a lot of trouble finding anything about this on the Net, but of course the words involved are not exactly unique! Please be gentle if there is something obvious I'm missing - I've put "elementary" as a tag by way of anticipating there is a clear answer. At least the statement is elementary!

Edit: I like all three answers for different reasons, and have voted up accordingly. None really answers my question, but that's because, upon further review, I think it's not well posed. After all, FTA is not an axiom like Postulate 5 (though of course one needs various axioms to prove it).

So maybe the answer about $a|bc$ is closest to what I was looking for, though as it happens I like to prove this first as well. Probably the best question would be how much one can prove in number theory without using the FTA. But that would be a different question! Thanks.

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When you remove the fundamental theorem of arithmetic, what are you leaving? (An axiomatic description of the integers themselves? Axioms for a commutative ring?) –  Qiaochu Yuan Jan 28 '11 at 21:16

4 Answers 4

Such a kind of question is the central concern of the field of mathematics known as Reverse Mathematics. The goal of the subject is to find exactly which axioms are needed to prove which theorems, over a very weak base theory, and the project has been completed for huge parts of classical mathematics. There is a particularly good book by Stephen Simpson on the topic.

One surprising outcome is that numerous classical theorems have turned out to be equivalent to each other, grouped in a comparatively small number of equivalence classes. Follow the link above for information about the five principal theories.

That said, I'm not sure that the results of reverse mathematics will actually provide you the answer you seek in the case of FTA. Most of the focus for reverse mathematics has been on theorems that are formalized in second-order number theory, and it seems that FTA is simply too weak to apply the main machinery of reverse mathematics, since it appears to be already provably in the standard base theory $RCA_0$.

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The Euler product for the Riemann zeta-function is equivalent to unique factorization in the integers.

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At least one textbook reserves the title Fundamental Theorem of Arithmetic for the one that says that if $a$ divides $bc$ and is relatively prime to $b$ then it divides $c$. I think you'll find it's easy to deduce either one of this theorem and unique factorization from the other.

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Pete Clark has an expository paper that is relevant to your question (though at the time of this writing, I can't seem to get through to the UGA website). In the particular case of FTA, I think it's more fruitful to ask for conditions on a commutative ring that imply (or are equivalent to) unique factorization, rather than to ask the reverse-mathematical question of the logical strength of FTA.

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Yes, I agree that this is probably what the OP wants. The reverse mathematics approach would be appropriate if the OP was specifically interested in that property about the actual integers, rather than the question of what one needs to get that property in a commutative ring. –  Joel David Hamkins Jan 31 '11 at 21:17

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