How Math's Most Famous Proof Nearly Broke | Hacker News Reader

How Math's Most Famous Proof Nearly Broke

nautil.us

113 pointskeehun17 comments

From the article: "Cauchy and Lamé hung their proofs on the tacit assumption that complex numbers, like real numbers, can be factored into a unique set of primes. The real number 6, for example, always equals 2 x 3. Aside from reordering the factors (3 x 2), no other product will work. But to Cauchy and Lamé’s embarrassment, their German contemporary Ernst Kummer showed that certain complex numbers can be split into prime factors in more than one way."

Well, there's also 6 = 1.5 x 4, etc. It's just in the context of the whole numbers (i.e., integers) that 6 has specifically the prime factorization 2 x 3. And depending on what your notion of a whole complex number is, they have unique prime factorization as well. For example, the Gaussian integers (complex numbers of the form a + b * i, where a and b are ordinary integers) have unique prime factorization.

The Gaussian integers are the n = 4 case of "Consider the complex numbers generated by adding, subtracting, and multiplying copies of a primitive n-th root of unity"; Lamé's mistake, pointed out and built upon by Kummer, was in assuming that the same property of unique factorization into irreducible elements would continue to hold even if n here was replaced by any prime (this assumption fails for the first time at n = 23).

Yes, the article did a vast oversimplification there.

In particular, Cauchy, Lamé, and Kummer were working with rings of algebraic integers[1].

One simple example is the ring denoted Z[sqrt(-5)], which is the set of all complex numbers whose real part is an integer and whose imaginary part is an integer multiple of sqrt(5), ie

Z[sqrt(-5)] = { a + b * sqrt(-5) | a,b are integers }

This ring has a lot of structure in common with the integers, but it does not have unique factorization. In particular:

    6 = 2 * 3 = (1 + sqrt(-5)) * (1 - sqrt(-5))

and each of 2, 3, 1 + sqrt(-5) and 1 - sqrt(-5) are irreducible in Z[sqrt(-5)] (ie they don't have any other factors other than 1 or -1). This is in opposition to what happens in the integers, namely unique factorization of any integer (other than 0, 1, or -1) into primes.

The failed proof mentioned above hinged on the following assumption: if p is a prime and

    r_p = cos( 2 * pi / p) + i * sin( 2 * pi / p)

is a primitive p^th root of unity, then the ring of algebraic integers

    Z[r_p] = {a_0 + a_1 * r_p + a_2 * (r_p)^2 + ... + a_(p-1) * (r_p)^(p-1) | a_0, a_1, ... , a_(p-1) are integers}

has unique factorization[2].

It turns out that Z[r_p] has unique factorization if p is a regular prime[3]. However, this is not the case for non-regular primes. Thus, Fermat's Last Theorem wasn't settled in the 19th century.

On the bright side, however, these failed attempts contributed to renewed interest in algebraic number theory.

[1]: http://en.wikipedia.org/wiki/Algebraic_integer

[2]: http://people.math.gatech.edu/~jrabinoff6/mathcamp/lectures....

[3]: http://en.wikipedia.org/wiki/Regular_prime

> Well, there's also 6 = 1.5 x 4, etc. It's just in the context of the whole numbers (i.e., integers) that 6 has specifically the prime factorization 2 x 3.

Not quite. You can easily extend prime factorization to rational numbers by allowing negative as well as nonnegative exponents for primes. The prime factorization for 6 (and any other rational) will still be unique.

If you instead decided to represent the factorization of a rational as a list of factors for the numerator and another list for the denominator, your factorizations would stop being unique... but the first way works just as well.

What you say is correct, interpreting "primes" as meaning "the values 2, 3, 5, 7, 11, etc.". Every positive rational is essentially uniquely a product of finitely many of these values raised to integer powers.

That having been said, what is it that makes these particular values "prime"? It's that they cannot be (nontrivially) decomposed into further factors. But in the rationals, they can be! 2 = 1.6 * 1.25, 3 = 1.6 * 1.875, etc. The inability to further factor 2, 3, 5, 7, 11, etc., which causes us to single them out as "prime" in the first place, is only on an account of factorization into whole numbers excluding such rational factorizations.

So it's in that context that my comment was to be understood: the fact that 6 specifically has the irreducible factorization 2 x 3 is not universally true, but only true in particular contexts, of which the most common would be the integers. In the mentioned context of the reals, or even just the rationals, we could instead note that 6 (or anything, really) could continue being divided indefinitely, thus having no particular "prime factorization" in the sense of a factorization which cannot be further refined.

The article is a very interesting popular account of the Fermat-Wiles Theorem and the discovery of the flaw in Wiles's first attempted proof of the theorem. I was glad to see that the article makes the timeline of Fermat's writings on the topic sufficiently clear to support the historical statement that Fermat himself surely didn't have a proof for the full theorem: "Fermat himself had given a proof for n = 4." (Writing marginal notes in the book about number theory he was reading was something that Fermat did early in his amateur study of mathematics, while doing his professional work as a lawyer. Fermat later "published" many proofs in the manner of his era by writing letters to other scholars. Fermat's son published the marginal notations in an edition of the book Arithmetica published only after Fermat's death. If Fermat actually had a proof for the theorem known as his last theorem, he had plenty of opportunity in his lifetime to find other pieces of paper on which to write it down.) I recall that the current textbook Mathematics and Its History by John Stillwell[1] reviews the history of this topic pretty well.

[1] http://www.amazon.com/Mathematics-Its-History-Undergraduate-...

Simon Singh's excellent book also recounts this fairly remarkable story.

I was actually studying Maths at Cambridge when Wiles gave his seminars. People were buzzing with excitement after the first one.

For those in the UK, BBC iPlayer has an old Horizon doc on this made shortly after the theorem was fixed currently. It's pretty good, and even has the old "lingering shot until the subject starts crying," except the subject is Wiles, getting emotional about his proof, and it's quite moving

I presume that you're referring to Wiles' statement to the effect that, "Proving Fermat's Last Theorem is probably the biggest thing that I'll ever do." As a young student, it made a big impression. Eighteen years later, as a professional scientist, it's even more poignant.

Found it in a NOVA transcript [1]:

"At the beginning of September, I was sitting here at this desk, when suddenly, totally unexpectedly, I had this incredible revelation. It was the most—the most important moment of my working life. Nothing I ever do again will. . . I'm sorry."

[1] http://www.pbs.org/wgbh/nova/transcripts/2414proof.html

Link to the documentary for those in the UK: http://www.bbc.co.uk/iplayer/episode/b0074rxx/horizon-199519...

The problem with all these proofs is that it'd take someone who already has a Math PhD several years of full-time effort to understand the proof.

Because these famous problems were solved, there is now increased demand for Math PhDs with that specialty.

How do you know if the proof is valid, or if they're just scratching each others' backs and pretending they're all brilliant? You might say "Wait, a fraud that big couldn't happen." Just like one of the biggest investment funds couldn't be a Ponzi Scam? (Madoff)

I.e., the few Mathematicians who know enough to be able to check the full proof might be doing the equivalent of this guy:

http://nymag.com/scienceofus/2015/05/how-a-grad-student-unco...

This isn't really a comparable example. We are talking about the most famous recent piece of math that made headlines all over the world first when the proof was published, again when issues were found with the proof, and again when those issues were addressed. Vs a "world famous in Poland" study in what not even barely qualifies as a science.

No one in bleeding edge science can fully grasp bleeding edge work outside their specialty, but high profIle stuff like this gets a lot of attention.

The only (gaping) hole in your conspiracy is that you get points for proving anyone wrong in academia (even yourself). A conspiracy about mathematic proofs sounds about as ridiculous as an honest segment of The O'Reilly Factor.

Yes, they might.

But then this gets published and other people could point their mistakes. And then it's not only about what the authors got wrong, but also what the reviewers let slip.

And while the number of mathematicians that can understand the proof is low, it is > 1

> How do you know if the proof is valid, or if they're just scratching each others' backs and pretending they're all brilliant?

(Professional mathematician here)

Such a conspiracy could never happen. Mathematicians are, in general, scrupulously honest to a fault. There are exceptions of course, but such a scam would need everybody's cooperation.

Moreover, if you have tenure then you would have no incentive to participate in such a fraud (even to the extent of keeping quiet about it). Once you get tenure you are basically working for pride and for the sheer joy of solving problems, and maintaining a big lie would do nothing for either.

I would like to both broaden and qualify that statement: scientists are, in general, scrupulously honest where it concerns their work, probably because their work is about 'truth'. Nevertheless, scientists are still people and many people lie and cheat when it suits them.

I still find Beal's conjecture very interesting. FLT follows quite trivially from it.

http://en.wikipedia.org/wiki/Beal%27s_conjecture

http://sumnotes.com/pages/a-simple-proof.html

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