
A very old problem turns 20 - ColinWright
http://plus.maths.org/content/very-old-problem-turns-20
======
nicholassmith
The history of the Theorem and the attempts to crack it, and how Wiles finally
got there is fantastic, and if you've not come across it in detail then I'd
definitely recommend finding out more about it.

If anyone is looking for a good book about the history of Fermat's Last
Theorem I can thoroughly recommend Simon Singh's book about it [1], which is
both accessible for those without a strong maths background (like myself) and
paints a wonderful history of mathematics as well. In places it's more
entertaining than fiction.

[1] [http://www.amazon.co.uk/Fermats-Last-Theorem-confounded-
grea...](http://www.amazon.co.uk/Fermats-Last-Theorem-confounded-
greatest/dp/1841157910)

~~~
dagw
Great book, no matter your level of math knowledge or interest.

I'm an all round science nerd with a MSc in maths, my wife is an architect
with a background in art history and graphic design and a aversion to all
things mathematical, yet we both really liked book.

~~~
dtparr
Somewhat off topic, but I was somewhat surprised that architect is a viable
career for someone with an aversion to all things mathematical. Does she still
need math, she just doesn't enjoy it, or does the modeling software these days
take care of all the calculations necessary?

~~~
dagw
The role of the architect in Sweden is very different from the role of the
architect in North America. In Sweden an architect deals with the planning and
design phase and is not involved with the construction phase and has no
practical or legal responsibilities with regards to the actual building
process. All that is handled by a construction engineer.

~~~
dtparr
Interesting. I would have thought at least some math would be required during
the planning/design phase to ensure the design is physically stable (e.g. load
calculations to ensure spans are supported well enough). Thanks for the info.

~~~
dagw
Most houses are designed with standard, well understood materials and well
within industry standard tolerances, so there is no need to do any heavy duty
load calculations at every step. 97% of the time plugging some numbers into
the standard formulas is plenty enough for a sanity check. If you're doing
something really weird you sit down in collaboration with your engineers.

------
tokenadult
Some comments ask whether Pierre de Fermat may have had a proof of his
conjecture (which I now call the Fermat-Wiles Theorem) using the mathematics
of his day. Agreeing with Colin, on historical grounds, I think Fermat did NOT
have a proof of his famous conjecture.

The historical reasoning works like this. In Fermat's day, mathematical
"publishing" was not submission of articles to peer-reviewed journals, but
people interested in mathematics sending personal letters to one another. (One
of the first mathematical journals, if I remember correctly, began as a
collection of mathematicians' letters to one another, something like turning
individual emails into an email list.) Fermat was a lawyer by occupation, and
did mathematics in his free time. He corresponded with other mathematicians of
his era, including Pascal. In all the surviving correspondence Fermat had with
other mathematicians, there is no trace of a proof of "Fermat's Last Theorem,"
even though, as Colin pointed out, he wrote about simpler cases of that
theorem.

The real historical reasoning comes in by considering dates of the primary
sources. Fermat's note to himself in the margin of the printed Latin
translation of Diophantus was written EARLY in Fermat's investigation of
number theory. He read that book, taking notes in the margins as he read, and
then he wrote for years afterward about interesting number theory problems.
The book with Fermat's annotations was published after Fermat's death by
Fermat's son, and that is when the marginal note became known to the world,
but Fermat had plenty of time--and plenty of pieces of paper with wider
margins--to write out a complete proof of his conjecture if he actually had
one. So as essentially all mathematicians have concluded on mathematical
grounds, and as all historians have concluded on historical grounds, most
likely Fermat briefly thought he had a complete proof of his conjecture, which
he then worked on, discovering a mistake in his reasoning. He proved a simpler
case of the theorem, died, and then when his notes were published, other
mathematicians worked on the full conjecture for 350 years before Wiles found
the complete proof, based on advanced research by other mathematicians still
alive today.

Google Books links to the historical background from mathematicians who write
good popular books about mathematics:

[http://books.google.com/books?id=I-RSVN6TjXsC&pg=PT95](http://books.google.com/books?id=I-RSVN6TjXsC&pg=PT95)

[http://books.google.com/books?id=ZhK73qVgOw0C&pg=PA265](http://books.google.com/books?id=ZhK73qVgOw0C&pg=PA265)

------
chewxy
Fun fact: In Star Trek the Next Generation, Picard mentions that Fermat's Last
Theorem had gone unsolved for 800 years.

5 years after that episode came out, Wiles showed his proof.

~~~
diminish
in Picard's parallel universe Wiles wasn't born.

~~~
ColinWright
See my answer elsewhere explaining why this is not true.

Here:
[https://news.ycombinator.com/item?id=5932237](https://news.ycombinator.com/item?id=5932237)

------
eksith
There's a lovely documentary from the BBC/Horizon on Andrew Wiles'
accomplishment. Also ran on PBS as the "The Proof".

[http://www.youtube.com/watch?v=7FnXgprKgSE](http://www.youtube.com/watch?v=7FnXgprKgSE)

It's a very moving piece that puts a lot of depth behind the 2d scribbling on
paper that describes whole universes. Also a very humane look into the lives
of mathematicians.

Reminds me so much of how people obsessed with a solution will give up any
semblance of a social life to pursue it.

~~~
ibrow
Here is the book[1] that goes with that documentary. It is a really good read
and not too technical at all.

[1] [http://www.amazon.co.uk/Fermats-Last-Theorem-confounded-
grea...](http://www.amazon.co.uk/Fermats-Last-Theorem-confounded-
greatest/dp/1841157910/)

Edit: Just seen nicholassmith posted exactly the same link. Apologies for
duplication

------
hmexx
I'm not a mathematician, but I find it unsatisfying that the solution to such
a simply stated theorem requires 100s of pages of mathematics.

Would have been cooler if the proof was extremely simple to write out, but
required outside-the-box thought process that evaded thousands of humans for
centuries.

I hope the same thing does not happen to physics. Discoveries that lead to
simple equations like e=mc^2 are so cool!

~~~
fractalsea
What I find interesting is that these proofs _aren 't_ one liners. To me, I
intuitively presume that mathematical proofs should be one liners because they
are explaining fundamental truths of nature; this universe and beyond.

Having said that, it's quite possible that a more "elegant" proof will be
discovered in the future. Maybe it requires further thinking outside the box
-- it will more likely require new mathematics.

If you want _really_ non-elegant proofs, take a look at computer-assisted
proofs (e.g. proof of the four color theorem). These make the subject of this
thread seem extremely elegant!

~~~
anonymous
It can be a one-liner, but you'll need more language. That is, you'll need
terms that encode mathematical truths that currently can only be expressed as
several pages.

Think of this - I can use the number two and refer to it with just a single
character - 2. However, properly stating all its properties and what it is
starting from just the axioms of set theory, I'll need quite a lot of pages.

~~~
VLM
Maybe a good way to resonate the situation with HN would be you wanna store
the number 2. Well, if you only know and are permitted to use classic IEEE 754
floating point then this is going to be a really long story. But once someone
"invents" binary integers and its widely accepted that you're "allowed" to
assume everyone understands them, then to store the number 2, you just squirt
out a 8-bit binary integer word 00000010 and call it good.

And that's how "lets store the number 2" goes from 1000 pages and ten hours of
lecture to explain to CS grads, to some 5 minute Kahn academy video that any
goof off the street can more or less understand.

There is no proof that any simpler explanation exists. Its quite possible no
one will ever teach the proof of FLT to grade school kids. But if it ever
happens it'll be like the analogy above.

------
andrewingram
I've occasionally caught myself wondering whether there is a completely
different proof that could have been discovered using 17th century
mathematics. Or is it the case that the maths used by Andrew Wiles would have
been understandable to Fermat?

Namely, I'm wondering if Fermat was actually lying and he just got lucky that
a proof was eventually found.

~~~
ColinWright
It is almost certain that Fermat did not have a proof. He did have
(effectively) a proof for the case n=4, but it seems he worked for some time
without success on a proof for n=3. It seems unlikely he would do so if he
really did have a general proof.

I'm told that if all rings of the form _a+b.sqrt(-p)_ are unique factorization
domains then there is a fairly simple proof[0], and it's plausible that Fermat
momentarily forgot that this was not in fact the case. If he found the "proof"
that uses this "fact" he may have got carried away and written his marginal
note, then moments later done a face-palm, and not bothered to erase or
correct that note.

[0] I do not speak from personal knowledge, and this may have been distorted.
I believe the essence to be correct.

~~~
wging
>I'm told that if all rings of the form a+b.sqrt(-p) are unique factorization
domains then there is a fairly simple proof

I just can't resist showing why this is false. It's actually pretty simple.

Take p=5. Then our 'integers' in this ring are of the form a + b\\*sqrt(-5)
for integers a, b. Taking b = 0 we see that all the normal integers are part
of this ring. So consider how 6 factorizes.

On the one hand we have 6 = 2 × 3... uncontroversial. But... due to the
existence of new numbers in this expanded ring, we also get 6 = (1 + sqrt(-5))
× (1 - sqrt(-5)) . These are evidently two different factorizations... it
suffices to show that both are prime factorizations under the modified
definition of 'prime' we get when we allow factors of (a + b × sqrt(-5)). You
can get there by noticing that any number of the form a + b × sqrt(-5) that
factors into smaller such numbers must either be totally real (b=0), be
totally imaginary (a=0), or have a larger absolute value than any number that
could possibly multiply together with another number > 1 to get you 1 +/\-
sqrt(-5). This line of reasoning can show that 2, 3, (1 + sqrt(-5)), and (1 -
sqrt(-5)) are all primes in this ring.

~~~
ColinWright
So you have proven that a ring of that form is not necessarily a UFD by giving
a specific case where it isn't, and proving it.

The real point is that assuming rings of that form _are_ UFDs gives a
relatively easy proof of FLT.

