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nt.quantity principle – analytic equivalences for FTA
I await this is not a dilatory query…
It’s well-known that (within the presence of assorted different axioms), Euclid’s posit 5 (‘parallel axiom’) is equal to the Pythagorean Theorem. That is, occupy Pyth. Thm. is correct with out posit 5, and also you purchase the ‘parallel axiom’ as a theorem.
My query: Are there well-known (or not-so-well-known) theorems/properties of the ring of integers that are equal to the Fundamental Theorem of Arithmetic on this route? That is, issues which aren’t simply penalties of it, however suggest it.
I maintain had loads of wretchedness discovering something about this on the bag, however of passage the phrases will not be precisely unique! delectation breathe light if there’s something patent I’m lacking – I’ve put “elementary” as a tag by route of anticipating there’s a limpid retort. At least the assertion is elementary!
Edit: I affection all three solutions for various causes, and maintain voted up accordingly. None actually solutions my query, however that is as a result of, upon additional evaluation, I arbitrator it is not effectively posed. After all, FTA just isn’t an axiom affection posit 5 (although of passage one wants numerous axioms to show it).
So possibly the retort about $a|bc$ is closest to what I used to be in search of, although because it occurs I affection to show this primary as effectively. Probably the perfect query would breathe how mighty one can show in quantity principle with out utilizing the FTA. But that might breathe a special query! Thanks.
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