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).
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
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.
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.
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.
> 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?
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!
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.
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).
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.
What I find interesting in this show is that he took a hiatus from his profesorship to spend the time to find the proof. I think it was years. Also, he found the proof by providing a different conjecture(is that right) to solve Fermat.
Higher mathematics is just a different world to those of us who haven't even a clue how that world functions.
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. :-)
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).
The BBC did an episode of Horizon on it.
https://www.bbc.co.uk/iplayer/episode/b0074rxx
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
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.
It does. That’s all part of the drama.
One of the best books I’ve ever read.
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.
Would it be good on audio?
Maybe. Most of it is just good storytelling. There are some light equations and graphs occasionally.
It does! Very dramatic part of the book.
This is the most offensively-themed serious site I have ever seen.
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.
It's fast, legible, dense and ad-free. Can't get any better than this.
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.
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.
That would be pretty anachronistic; jQuery didn't come around until six years later.
Well since jQuery was 6 years in the future, it is understandable they didn’t use it.
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.
On purely aesthetic grounds I'll take this over another substack-like site any day of the week.
> 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?
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!
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.
efficient mathematics hypothesis? What's that?
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.
They're the same thing; knowledge markets are markets.
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...
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).
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.
What I find interesting in this show is that he took a hiatus from his profesorship to spend the time to find the proof. I think it was years. Also, he found the proof by providing a different conjecture(is that right) to solve Fermat.
Higher mathematics is just a different world to those of us who haven't even a clue how that world functions.
I have a truly marvellous comment on this, which this comment box is too small to contain.
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. :-)
I assume you meant Gabriel Lamé
From 2000.
Posting because he's retiring this year?
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.")
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.