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).
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.
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.
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.
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. :-)
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.
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!
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.
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.
> 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...
> 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).
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.
> 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?
albert_e | 6 hours ago
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).
chris_st | 6 hours ago
clarkeni | 6 hours ago
WD-42 | 5 hours ago
One of the best books I’ve ever read.
stogot | 5 hours ago
WD-42 | 5 hours ago
placebo | 5 hours ago
tonyedgecombe | 5 hours ago
https://www.bbc.co.uk/iplayer/episode/b0074rxx
matthewfelgate | 6 hours ago
HarHarVeryFunny | 5 hours ago
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.
alok-g | an hour ago
NetMageSCW | 21 minutes ago
fnands | 5 hours ago
Posting because he's retiring this year?
FeepingCreature | 5 hours ago
fnands | 5 hours ago
But agreed, that lime green is horrendous.
jebarker | 5 hours ago
energy123 | 4 hours ago
drivers99 | 3 hours ago
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.
libria | 2 hours ago
Table-based layout, font tags, map/area tags. Only thing missing is an unnecessarily imported jQuery.
smrq | an hour ago
NetMageSCW | 25 minutes ago
NiloCK | 5 hours ago
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!
clircle | 4 hours ago
fancyfredbot | 4 hours ago
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.
mathisfun123 | 4 hours ago
gosub100 | 2 hours ago
rheniumclass | an hour ago
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...
mathisfun123 | an hour ago
> 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).
danbruc | 3 hours ago
wiz21c | 2 hours ago
(the site is : "An ongoing multi-author open source project to formalise a proof of Fermat's Last Theorem in the Lean theorem prover.")
jfengel | 40 minutes ago
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.
bambax | 2 hours ago
> 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?