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I'm looking for a reference containing the following proof of Euclid's Lemma.

Recall the statement: Let $a,b$ be positive integers and let $p$ be a prime dividing $ab$. Then $p$ divides $a$ or $b$.

[I'm rewriting the proof taking into account Geoff Robinson's comments:]

Let $Q$ be the set of those quadruples $(p,c,a,b)$ of positive integers such that

  • $p$ is prime,

  • $ab=pc$,

  • $p$ divides neither $a$ nor $b$.

Assume by contradiction that $Q$ is nonempty and let $(p,c,a,b)$ be its least element with respect to the lexicographical ordering. We clearly have $c > 1$. We also have $c < p$ because otherwise $(p,c-a,a,b-p)$ would be in $Q$.

[Edit (Aug. 4): Assuming $c\ge p$, we have $b > p$ (and thus $c > a$), for the contrary would imply $a\le b < p$, and thus $ab < p^2\le pc=ab$. (Thank you to Bill Dubuque!)]

If $q$ is a prime factor of $c$, then $(p,c/q,a/q,b)$ or $(p,c/q,a,b/q)$ is in $Q$, a contradiction.

EDIT (July 24). Thank you to Martin Sleziak for his comments. I'll look quietly at his links, and make perhaps a new edit. In the meantime here is the argument used by Gauss in his Disquisitiones Arithmeticae, Art. 13. [I should have studied this Article more seriously before posting my question. My apologies.]

Let $p$ be a prime and $a$ a positive integer not divisible by $p$. Assume by contradiction that the set $B$ of all positive integers $x$ such that $p$ divides $ax$, but doesn't divide $x$, is nonempty, and let $b$ be the minimum of $B$. The inequalities $1 < b < p$ are clear. There is a positive integer $m$ such that $mb < p < (m+1)b$, and one easily checks that $p-mb$ is less than $b$ and belongs to $B$, a contradiction. QED

Actually I like Gauss's argument much better than the one I gave above. But here is what puzzles me. Many Elementary Number Theory and Algebra textbooks prove the Fundamental Theorem of Arithmetic (or Euclid's Lemma) using only Peano like axioms, and in particular not using the construction of $\mathbb Z$ from $\mathbb N$. I find it's a very good idea to do so. [I agree that the construction of $\mathbb Z$ is of paramount importance, but I believe, it's also important to realize that the Fundamental Theorem doesn't rely on it.] The trouble is that these proofs, I feel, tend to be much more complicated than the above argument of Gauss. [Thank you for correcting me if I'm wrong.]

Pierre-Yves Gaillard
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  • @Geoff Robinson - Thanks for your comment. I tried to make things less cryptic by adding an edit. – Pierre-Yves Gaillard Jul 23 '11 at 11:27
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    Yes, thanks, that clarifies it. I slightly re-edited my comment, since I was cut off last time. I suppose your argument is somewhat in the tradition of Fermat- if there is a solution, there is a minimal one, but then you can produce a smaller one. You could almost says yours is a proof "by induction on the prime"- take p as the minimal prime for which the Lemma fails (note that p =2 is OK!). – Geoff Robinson Jul 23 '11 at 15:15
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    Trouble editing, but (for the record), here's what my first comment was meant to be:"Your proof is a little cryptic, but the idea is very nice. I suppose what you meant was that $a <p$ and $b <p$, and that $c$ has no common factor with $a$ or $b$." – Geoff Robinson Jul 23 '11 at 15:25
  • @Geoff Robinson - Again, thank you very much! – Pierre-Yves Gaillard Jul 23 '11 at 15:34
  • BTW I was about to suggest to ask the editor which contributed this proof to wikipedia article about the source, but it seems that this edit was anonymous http://en.wikipedia.org/w/index.php?title=Euclid%27s_lemma&action=historysubmit&diff=335989446&oldid=332202525 (I guess if it were your edit, you would mention the wikipedia article.) – Martin Sleziak Jul 24 '11 at 08:58
  • @Martin Sleziak - Thank you very much! I’ll take a close look at your links. (I didn’t write the Wikipedia edit.) – Pierre-Yves Gaillard Jul 24 '11 at 09:34
  • @Martin Sleziak - If you turn your comment into an answer, I’ll upvote it and accept it. – Pierre-Yves Gaillard Jul 25 '11 at 03:48
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    @Pierre-Yves: I added my comments as an answer. I think you might want to wait with accepting the answer - perhaps someone will have something interesting to say about your last paragraph. – Martin Sleziak Jul 25 '11 at 07:05
  • @Pierre-Yves I hope to soon post a (long) reply. But before doing so, could you please clarify the nature of your query. Do you simply seek references, or are you interested in more, e.g. attempting to classify these elementary proofs. Why do you think it is a very good idea to restrict the proof to $\mathbb N$ vs. $\mathbb Z$ ? What sort of proofs have you seen that are much more complicated than that of Gauss? Are you aware of the elementary proofs of unique factorization by Kronecker, Zermelo, Hasse, Lindemann and Klappauf? (references and proofs will be in my reply). – Bill Dubuque Aug 04 '11 at 13:22
  • @Bill Dubuque - Thank you very much in advance! - N vs. Z: I didn't explain it, but the question I really asked myself is this. Suppose you want to add to Bourbaki's treatise a proof of the FTA which would come as early as possible in the treatise, and would be as simple as possible. Which proof would you give? Bourbaki introduces N very soon, and Z much, much later. - The elementary proofs: Compared to Gauss's proof, I find them more complicated but not more elementary. I'm must be missing something. - The interpretation of the question: I have no restriction whatsoever. – Pierre-Yves Gaillard Aug 04 '11 at 13:59
  • @Bill Dubuque - Cont. Please feel free to be as long as you wish. - Going back to the N vs. Z: a variation of the question could be: how to derive the FTA from Peano’s axioms as swiftly as possible. More generally, I think we all agree that the FTA doesn’t depend on the construction of Z. – Pierre-Yves Gaillard Aug 04 '11 at 14:04
  • @Pierre-Yves In my youth I once classified all of these "elementary" proofs. I convinced myself that they are all essentially gcd/ideal-theoretic, in the sense that they are specializations of proofs that also work for Euclidean/gcd domains or PIDs. But, in my opinion it is best pedagogically to preserve this innate structure by abstracting out the descent into a Lemma, e.g. see my post here and see esp. its linked network of posts. By the way, precisely which way does your sketched first proof deduce that $c < p$? – Bill Dubuque Aug 04 '11 at 15:03
  • @Bill Dubuque - Thanks for pointing out this point which was unclear. I made an edit. I hope it’s now clear, and correct! I’ve taken a first look at your post. They seem great to me. You have obviously pushed the reflection much further than I. Again, I’m looking forward to your answer! – Pierre-Yves Gaillard Aug 04 '11 at 15:46
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    @Bill Dubuque - I was wondering: do you have an estimate of the total number of your posts (on the whole web)? – Pierre-Yves Gaillard Aug 04 '11 at 17:47
  • I saw your now removed dénouement comment to my Grothendieck piece and I failed to answer in time because I was distracted by other matters. Thank you and I hope you didn't retract it for reasons of content. – t.b. Aug 04 '11 at 21:10
  • @Theo Buehler - Thanks! Of course I’m retracting nothing. I deleted the comment because I thought you had more important things to do than answering such comments. – Pierre-Yves Gaillard Aug 05 '11 at 01:03
  • @Pierre In proof 1, why cancel primes from $c$ vs. more simply from $b$? Do you exclude "ideal-theoretic" proofs and use of division vs. subtraction? If so, why accept Gauss's division-based proof? Also it's just a variant of the Euclidean algorithm: $(p,b) \to (p,b-p)$ if $ b>p$ else $(p, p\ mod\ b)$, eventually reaching $(p,1)$ hence $p|a\cdot 1$. Essentially it uses gcds: $p|ab:$ so $p|a(b,p)$, and $(b,p) = 1$ is computed by a variant of the euclidean algorithm with one argument fixed prime. It can be used to compute inverses. – Bill Dubuque Aug 09 '11 at 04:40
  • @Bill: I agree that proof 1 is silly. I explicitly apologized for it in the question. I don't want to avoid subtraction. What I want to avoid is negative integers. I don't "exclude" ideal-theoretic proofs. Many mathematicians have produced proofs of the FTA avoiding negative integers. I presume (correct me if I'm wrong) that this applies to the proofs by Kronecker, Zermelo, Hasse, Lindemann and Klappauf that you mention. I asked myself a very precise (and, I agree, very artificial) question, which was: ... – Pierre-Yves Gaillard Aug 09 '11 at 06:12
  • @Bill: … Suppose you had to add to Section III.5 of Bourbaki's set theory book an exercise aimed at proving the FTA. What would you do? Bourbaki proves in this section the existence of the euclidean division in N, so it's available. At that stage, Bourbaki's reader doesn't even know what a magma is. The 4 operations are (partially) defined on cardinals. N is just the set of finite cardinals. – Pierre-Yves Gaillard Aug 09 '11 at 06:13
  • @Pierre I'm trying to understand your definition of "simpler". In a comment to a deleted answer you wrote "I found some proofs avoiding negative integers and/or euclidean division, but they tend to be less simple". Hence my query above as to whether you prefer proofs using subtraction vs. division, since this seems to contradict your preference for Gauss' proof - which does use division. Almost all integers proofs I know also extend easily to naturals, so that's usually not a problem. This leaves only the preference for (iterated) subtraction vs. division. – Bill Dubuque Aug 09 '11 at 12:17
  • @Bill: Perhaps I didn't express myself very well, so let me try again. As far as I can see Gauss's proof doesn't use more than, say, this proof. But I find Gauss's proof much easier to read. I agree that Gauss's proof uses division, but I'm ready to use all that's been proved in the Bourbaki's set theory book. I also agree with the sentence starting with "Almost all". My personal preferences don't really matter. I hope you're still planning to post an answer, and I hope this answer will reflect as faithfully as possible your preferences! – Pierre-Yves Gaillard Aug 09 '11 at 12:43
  • @Pierre The linked proof is essentially Zermelo's, though not expressed as concisely or as conceptually as could be. If you "inline" Gauss's proof of Euclid's Lemma into a complete proof of unique factorization it will be roughly the same length as Zermelo's, e.g. see the other proofs mentioned by Wolke in his intro to Zermelo's paper in his Collected Works (which has some history). The earliest is a proof of Kronecker in his Vorlesungen, if memory serves correct. Are you already familiar with these other proofs? – Bill Dubuque Aug 09 '11 at 13:08
  • No, I’m not familiar with them. I’ve only looked at the proofs in Hardy and Wright, and in Davenport. I’m far from knowing the subject as well as you! I think it would be great if you included Zermelo’s proof in your answer. The ground rule of my MSE question was in my mind to avoid negative numbers. Among the few proofs fitting this bill I’ve seen, Gauss’s is the one I found most appealing. That’s all I’m saying. I’ll probably change my mind after your answer. – Pierre-Yves Gaillard Aug 09 '11 at 13:21
  • I see. Before I presumed that you had already seen these proofs but that they didn't satisfy all of your constraints. I plan to mention a few proofs in my answer. I thought it best to first clarify your query, in order to be able to provide a more helpful reply. Discerning the nuances of another's notion of "simpler" is never an easy task. Alas, I haven't yet found enough spare time to finish my reply. – Bill Dubuque Aug 09 '11 at 13:31
  • @Bill: Great! Thank you very much! Please explain what you think is important, and don’t pay too much attention to what I say! – Pierre-Yves Gaillard Aug 09 '11 at 13:44
  • @Pierre See also my answers here.. I hope to elaborate more later. – Bill Dubuque Mar 07 '12 at 21:20

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I tried to google for "euclid's lemma" "divides neither". It seems that wikipedia provides similar proof in the part "Alternate proof without Bézout's identity", but it does not give a reference. Also this paper contains a proof similar to yours, but not exactly the same.

I was about to suggest to ask the editor which contributed this proof to wikipedia article about the source, but it seems that this edit was anonymous. I think that if you find a textbook containing this proof, it would be a good idea to add it as a reference to wikipedia article.

Martin Sleziak
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