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The Fundamental Theorem of Arithmetic says that every positive integer can be expressed uniquely as a product of primes. Although this theorem is famous and has been known for a long time, it is not trivial, and is not as simple to prove as the fact that every positive integer can be written in at least one way as a product of primes.

There's a cute example that I think does a good job of explaining to students (and to experienced mathematicians, for that matter) why the Fundamental Theorem of Arithmetic isn't obvious. I saw it somewhere when I was an undergraduate, but I can't remember where.

That example is: Suppose our number system consists of all even positive integers, and we call such a number "prome" if it cannot be written as a product of two smaller numbers in the set. Then:

2 is prome

4 is not prome (4 = 2*2)

6 is prome (because 1*6 and 2*3 are not factorizations into even integers)

8 is not prome (8 = 2*4)

10 is prome (because 1*10 and 2*5 are not factorizations into even integers)

12 is not prome (12 = 2*6)

We can then notice that 36 can be written as a product of "promes" in two different ways: 2*18 and 6*6.

My question is: Where did I see this argument? I believe I may have read it in a textbook in the library of my undergraduate institution, which would make it no later than 1995. (The word "prome" was used in the document I read; it's not my paraphrase.)

idmercer
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    I found, with Google, Elementary Number Theory by Underwood Dudley uses the word "prome" to describe a similar example of multiplicatively irreducible elements in the set $4\mathbb N + 1$ - maybe that's what you're thinking of? – Milo Brandt Oct 07 '19 at 16:22
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    Joseph H Silverman, in Chapter 7 of A Friendly Introduction to Number Theory, makes a similar presentation, calling the system of even numbers "the $\mathbb E$-zone." – Keith Backman Oct 07 '19 at 16:31
  • Interesting, and thanks to you both! It's possible that I conflated Dudley's discussion (using the word "prome") with the even number example described by Silverman (and/or possibly others). – idmercer Oct 07 '19 at 16:35
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    See also the Hilbert naturals $,1+4\Bbb N,,$ e.g. here and here, both of which include further related discussion on this topic. – Bill Dubuque Oct 07 '19 at 16:58
  • What does prome mean? I couldn't find it related to math in a google search. – poetasis Oct 07 '19 at 17:33
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    @poetasis From the post: "Suppose our number system consists of all even positive integers, and we call such a number 'prome' if it cannot be written as a product of two smaller numbers in the set. " – Kevin Long Oct 07 '19 at 19:17
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    @poetasis 'prome' is not a standard math term. It's a definition just given (by Dudley, apparently) for pedagogical reasons. – Jair Taylor Oct 07 '19 at 22:15

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