Is √17 irrational?
A thread
I learned recently of a cute, apparently open, problem that I think is interesting and merits some mention.
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A thread
I learned recently of a cute, apparently open, problem that I think is interesting and merits some mention.
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The standard even-odd proof of the irrationality of √2 goes as follows: Suppose p/q=√2, where p,q are positive integers. We may further assume that they are relatively prime. We have p^2=2 q^2, and a brief analysis by cases shows that both p,q should be even, contradiction.
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Once, teaching an introduction to proofs class, I spent some time analyzing this proof, and then asked students to attempt variations of it to prove, say, that √5 is irrational.
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Naively, I expected them to say that if p^2=5q^2 then p is a multiple of 5 and then so is q, and we have a contradiction. After all, I had spent what I thought was a reasonable time emphasizing the properties of 2 that made the proof work in the √2 even-odd analysis.
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Instead, some students argued by considering again an even-odd analysis. The whole thing seemed crazy, but enough of them did it that instead became suspicious, and I quickly found a book that irresponsibly argued this way. (And that the students had... "consulted"...).
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When preparing this thread, I looked at the book that I remembered as the culprit, and could not find this there, so who knows. Anyway, the proof ran something like the following:
If p^2=5 q^2 then p cannot be even or else so would be q, and vice versa, so they are both odd,
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If p^2=5 q^2 then p cannot be even or else so would be q, and vice versa, so they are both odd,
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say p=2k+1, q=2s+1, and we have
4k^2+4k+1 = 20s^2 + 20s +5,
or k^2+k = 5s^2+5s +1,
or 1 = k(k+1) - 5s(s+1),
so 1 is the difference of two even numbers, a contradiction.
As I said, crazy.
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4k^2+4k+1 = 20s^2 + 20s +5,
or k^2+k = 5s^2+5s +1,
or 1 = k(k+1) - 5s(s+1),
so 1 is the difference of two even numbers, a contradiction.
As I said, crazy.
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But the point is, of course, that one then gets curious. How far can this irresponsible way of arguing actually work? It turns out that it works to prove the irrationality of √3, √5, √6, √7, √8, √10, √11, √12, √13, √14 and √15.
I couldn't make it work for √17.
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I couldn't make it work for √17.
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Any argument "along the same lines" just seemed to balloon and go on forever.
Eventually, I found some references where other lunatics had attempted the same approach.
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Eventually, I found some references where other lunatics had attempted the same approach.
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Most notably, we have the following passage in the Theaetetus, by Plato.
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Now, why? Why would Theodorus stop at seventeen, one wonders.
The problem has been studied since: Can "the arithmetic of the even and the odd" proof the irrationality of numbers of the form (8k+1)2^n that are not already squares?
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The problem has been studied since: Can "the arithmetic of the even and the odd" proof the irrationality of numbers of the form (8k+1)2^n that are not already squares?
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These are the numbers for which the obvious even-odd analysis does not quite work.
The question has been studied by different authors. Part of the problem, of course, is to understand what precisely one is asking.
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The question has been studied by different authors. Part of the problem, of course, is to understand what precisely one is asking.
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A modern reference is
MR0416824 (54 #4893)
McCabe, Robert L.
Theodorus' irrationality proofs.
Math. Mag. 49 (1976), no. 4, 201–203.
doi.org/10.1080/002557…
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MR0416824 (54 #4893)
McCabe, Robert L.
Theodorus' irrationality proofs.
Math. Mag. 49 (1976), no. 4, 201–203.
doi.org/10.1080/002557…
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McCabe's paper is reviewed by Waterhouse, who does not think much of the problem: "he can find no such argument for 17. {None indeed can exist, as 17 is a square modulo 2^m for all m.}"
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I think instead of dismissing the question so quickly we see that the problem is not one of number theory but perhaps one of mathematical logic, trying to define precisely what the question is.
Victor Pambuccian worked on this recently.
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Victor Pambuccian worked on this recently.
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Pambuccian presents a formal weak theory of arithmetic that allows for even-odd reasoning, and shows that it cannot prove the irrationality of √17.
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But he does not seem satisfied with his theory actually capturing all that there is to the subject, because he revisits his axiomatization a bit later.
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And then Celia Schacht made an attempt at such a formalization as well.
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In his review of Schacht's paper for Mathematical Reviews, Pambuccian writes that the problem of the irrationality of √17 in Shacht's system is left open, but
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"What we do have for the first time ... is the right theory in which to prove the impossibility of an irrationality proof for √17. Its solution would finally provide a ... proof to the claim that one cannot prove by means of even and odd considerations that √17 is irrational."
Anyway, neat. A strange and in my opinion definitely interesting detour from what seemed like an utterly silly approach in an introductory class.
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