Solving Fermat: Andrew Wiles

Posted by 1970-01-01 1 day ago

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Comment by albert_e 22 hours ago

This was an excellent introduction to this topic:

https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem_(book) by Simon Singh

But I am not sure if the book covers the mistake and later correction. It has been more than a decade since I read the book (and became a fan of the author).

Comment by tonyedgecombe 21 hours ago

The BBC did an episode of Horizon on it.

https://www.bbc.co.uk/iplayer/episode/b0074rxx

Comment by cguess 10 hours ago

It's the same show, Nova just rebranded it in the US. BTW, here's a full video of the BBC version but available everywhere. https://www.dailymotion.com/video/x223gx8

Comment by chris_st 22 hours ago

I'll check that out - I loved his "The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography". Excellent introduction to cryptography.

Comment by WD-42 22 hours ago

It does. That’s all part of the drama.

One of the best books I’ve ever read.

Comment by placebo 21 hours ago

Agreed. I bought it at the airport for a transatlantic flight the year it came out and started reading before lift off. Completed it before landing. Best flight ever.

Comment by stogot 21 hours ago

Would it be good on audio?

Comment by WD-42 21 hours ago

Maybe. Most of it is just good storytelling. There are some light equations and graphs occasionally.

Comment by clarkeni 22 hours ago

It does! Very dramatic part of the book.

Comment by FeepingCreature 21 hours ago

This is the most offensively-themed serious site I have ever seen.

Comment by energy123 20 hours ago

It's fast, legible, dense and ad-free. Can't get any better than this.

Comment by fnands 21 hours ago

In their defense, this probably looked really cool in 2000, when this was published (or at least, last updated).

But agreed, that lime green is horrendous.

Comment by drivers99 19 hours ago

In Firefox, I didn't see a reader view available (I wonder what determines that), but I was able to right click, Inspect, change the <body bgcolor="#CCFF33" ...> background to #ffffff

Personally, I liked the design when zoomed in when combined with the rest of the page design. It reminds me of the 90s (Wired magazine, etc.). "Updated November 2000". That explains it.

Comment by libria 18 hours ago

The view-source is like a portal to Year 2000.

Table-based layout, font tags, map/area tags. Only thing missing is an unnecessarily imported jQuery.

Comment by smrq 17 hours ago

That would be pretty anachronistic; jQuery didn't come around until six years later.

Comment by NetMageSCW 16 hours ago

Well since jQuery was 6 years in the future, it is understandable they didn’t use it.

Comment by DStiego 12 hours ago

The background is so green that when I go look at a website with white background afterwards, the white will appear like red because the cells for green in my retina have been depleted.

Comment by jebarker 21 hours ago

On purely aesthetic grounds I'll take this over another substack-like site any day of the week.

Comment by bambax 18 hours ago

> NOVA: So Fermat's original proof is still out there somewhere.

> AW: I don't believe Fermat had a proof. I think he fooled himself into thinking he had a proof. But what has made this problem special for amateurs is that there's a tiny possibility that there does exist an elegant 17th-century proof.

Yes, it's generally accepted that Fermat didn't in fact have a proof, with the tools available to him at the time. But wouldn't it be cool to send some AI on this chase and see what comes back? Is anyone attempting this?

Comment by NiloCK 21 hours ago

Recent LLM dingers like the Jacobian Conjecture counterexample have challenged the efficient mathematics hypothesis. The JC counterexample was so small in degree and coefficient. It should have been a "low fruit" in the scheme of things, but alas, unpicked for 50+ years with considerable attention from good mathematicians.

With respect to FLT, my hopes have modestly increased that a truly marvelous demonstration of this proposition does in fact exist, that Fermat actually had it, and that it may someday be recovered!

edit: some emphasis on modest. But let me be romantic here!

Comment by danbruc 19 hours ago

There are 120 monomials of degree at most 7 in three variables. If we restrict the coefficients to the integers from -10 to +10, that makes 21^120 possible polynomials. And we need three of them, that makes 10^476. And we would still miss the specific counterexample because it includes a coefficient of 12 outside of our range. So I would say that you will never find this specific counterexample by chance and whether you could accidentally trip over any counterexample really depends on their density. And we have of course not addressed the question why you would search this specific region of the parameter space, why dimension 3, degree 7 and small integer coefficients? There might be good mathematical reason to look at this region, but it is probably non-trivial to even figure out where to look.

Comment by clircle 20 hours ago

efficient mathematics hypothesis? What's that?

Comment by fancyfredbot 20 hours ago

It's a pun on "efficient markets hypothesis".

The efficient markets hypothesis says the market prices incorporate all available information and there's no such thing as a cheap stock, so there's no easy money to be made by trading.

The math equivalent is presumably that all easy problems have been solved and all open problems should be very very hard. This has turned out not to be the case as LLMs have found simple counterexamples to long held conjectures.

Comment by mathisfun123 20 hours ago

They're the same thing; knowledge markets are markets.

Comment by gosub100 18 hours ago

Not really, the incentives for each are not binary values. There are bounties on the 10 millennium problems (that haven't been increased in 25 years), but there isn't much motivation beyond that. You get some street cred and your name in a prestigious journal. That's not-nothing but can't hold a candle to the potential fortunes of the market.

Comment by rheniumclass 17 hours ago

> There are bounties on the 10 millennium problems (that haven't been increased in 25 years) ...

Strictly speaking, if you constructively prove P=NP and the solver has reasonable time complexity, e.g. quadratic, it could make you incredibly rich. If you don't attract the wrong kind of attention from the world's intelligence agencies first...

Comment by mathisfun123 17 hours ago

i think you have no idea what you're saying...

> That's not-nothing but can't hold a candle to the potential fortunes of the market.

a market is where things are sold and bought. there are lots of markets which aren't as well capitalized as financial markets - your local farmer's market doesn't "hold a candle to the potential fortunes of the market" but it's still a market (and presumably the efficient market hypothesis still applies).

Comment by gosub100 1 hour ago

The markets arent efficient without enough participants. This is why the price of gasoline only fluctuates cents between competitors, but vehicle prices fluctuate thousands. It's a very simple concept that you aren't seeing.

The market for solving math problems is tiny compared to selling/ running algos for trading. You solve a pure math problem you might get accolades from some professors.

Comment by wiz21c 18 hours ago

I'm not a mathematician and AI doesn't answer very well. Could someone tell us how big an endeavour this is: https://github.com/ImperialCollegeLondon/FLT ?

(the site is : "An ongoing multi-author open source project to formalise a proof of Fermat's Last Theorem in the Lean theorem prover.")

Comment by jfengel 16 hours ago

Enormous.

Wiles' proof is 129 pages long, and builds on results that require a vast amount of infrastructure to define.

It's going to take dozens of person-years.

Comment by fnands 22 hours ago

From 2000.

Posting because he's retiring this year?

Comment by matthewfelgate 22 hours ago

I have a truly marvellous comment on this, which this comment box is too small to contain.

Comment by HarHarVeryFunny 21 hours ago

FWIW, it seems pretty clear that while Fermat briefly thought he had a proof, he fairly quickly realized it was flawed since he never mentioned this publicly and later went on to develop a proof just for the simple n=4 case.

In 1847 Gabriel Lami presented a claimed (simple) general proof to the French Academy of Sciences, only for the flawed assumption in it to be pointed out immediately at the end of his presentation! This may have been the same proof that Fermat had in mind.

Comment by alok-g 17 hours ago

Oh, so if he later published a proof for a special case (and presumably had more paper available than the book's margin :-) ), that's convinces me that he never had a correct proof. Also that he was smart enough to realize that but perhaps just forgot to or didn't erase his original margin note. :-)

Comment by NetMageSCW 16 hours ago

I assume you meant Gabriel Lamé