

How Math's Most Famous Proof Nearly Broke - keehun
http://nautil.us/issue/24/error/how-maths-most-famous-proof-nearly-broke

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Chinjut
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).

~~~
thaumasiotes
> 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.

~~~
Chinjut
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.

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tokenadult
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-...](http://www.amazon.com/Mathematics-Its-History-
Undergraduate-Texts/dp/144196052X/)

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moomin
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.

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gp7
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

~~~
ISL
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](http://www.pbs.org/wgbh/nova/transcripts/2414proof.html)

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fsk
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...](http://nymag.com/scienceofus/2015/05/how-a-grad-student-uncovered-a-
huge-fraud.html)

~~~
impendia
> 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.

~~~
Confusion
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.

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

[http://en.wikipedia.org/wiki/Beal%27s_conjecture](http://en.wikipedia.org/wiki/Beal%27s_conjecture)

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allthatglitters
[http://sumnotes.com/pages/a-simple-
proof.html](http://sumnotes.com/pages/a-simple-proof.html)

