Insights Fermat's Last Theorem

[fresh_42]

TL;DR

Why it took 350 years and a genius to prove Fermat's Last Theorem.

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation
$$
a^n+b^n=c^n
$$
has no solutions with positive integers if $n>2.$ It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy

"Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duas ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet."

"It is not possible, however, to decompose a cube into two cubes, or a biquadrate into two biquadrates, and in general, to decompose a power higher than the second into two powers with the same exponent: I have discovered a truly wonderful proof for this, but this margin is too narrow to grasp here."

which has been found after his death. Everybody understands the problem statement, however, it turned out to be extraordinarily difficult to solve, and nobody today assumes that Fermat had actually found a proof for the general case. For example,
$$
6^3+8^3=9^3-1,
$$
is almost a solution for the cubic case. Imagine how many natural numbers we have to rule out as possible solutions. Numbers we cannot even write down in a lifetime. Such considerations aren't even evidence in number theory since exceptions can always occur, but it shows the principal difficulty of any proof of negative results. Non-existence is hard to prove.

Both aspects of the story - the simplicity of the statement and Fermat's provocative remark - were a blessing as well as a curse. A blessing because, during the 350 years it took before Andrew Wiles and Richard Taylor succeeded in providing a complete proof in 1994, extraordinary developments in mathematics were initiated in the effort to prove the statement, especially in number theory, abstract algebra, class field theory, and the theory of elliptic curves. Its curse, however, lies in the fact that to this day, even after the more than 100-page solution to the problem, full of elaborate mathematics, laypeople still attempt to prove the theorem using simple means. It's safe to say that such attempts are doomed from the start. This article aims to shed light on why this is the case.

Continue reading ...

 

[jedishrfu]

What was Fermat's level of mathematical knowledge time?

Was it comparable to what Andrew Wiles knew?

I’ve read that Fermat would write to other mathematicians of his time, presenting a problem without providing his own solution. Hence, there are few known proofs done by him.

Additionally, mathematicians during his time were less rigorous than those of today's proof-oriented mathematics.

 

[Office_Shredder]

What was Fermat's level of mathematical knowledge time?

Was it comparable to what Andrew Wiles knew?

It seems exceedingly unlikely because most of the techniques Wiles used were from branches of math invented well after Fermat died.

 

[bob012345]

Obviously Fermat thought he had a proof. Are there good candidate “proofs” people think may have been what he did and what are the typical flaws?

 

[Pikkugnome]

It is unfortunate that you picked a near miss by one, since it has an obvious parity problem.

 

[jedishrfu]

But the example does illustrate how close you can get and still its not close enough to disprove the theorem.

A recollection by G.H. Hardy while visiting Ramanujan at a local hospital via a taxicab with a rather dull number. Ramanujan quipped, "Oh no, Prof. Hardy, it is a most interesting number as being the sum of two cubes in two different ways: 9^3 + 10^3 and 12^3 + 1^3."

Thereafter, 1729 was known as a taxicab number.

The significance of this story lies in Ramanujan's investigation of sums of cubes to challenge Fermat's theorem. Several examples of near misses were discovered in his notebooks, along with a scheme to generate even more.

Prof Littlewood, Prof Hardy's associate, once said that all positive integers were Ramanujan's personal friends.

https://www.americanscientist.org/article/never-a-dull-number

https://plus.maths.org/content/ramanujan

 

[martinbn]

Obviously Fermat thought he had a proof. Are there good candidate “proofs” people think may have been what he did and what are the typical flaws?

He had a proof for $n=4$. For the general case one possibility, but this only a guess there is no hint of what his marvellous proof might have been, is that you factorize, for which you need $p$th roots of unity, and work within the ring $\mathbb Z[\xi]$, and expect that it has unique factorization just like $\mathbb Z$.

 

[bob012345]

He had a proof for $n=4$. For the general case one possibility, but this only a guess there is no hint of what his marvellous proof might have been, is that you factorize, for which you need $p$th roots of unity, and work within the ring $\mathbb Z[\xi]$, and expect that it has unique factorization just like $\mathbb Z$.

Did he have rings?

 

[martinbn]

Did he have rings?

Of course not! Is this a serious question?

 

[pines-demon]

Did he have rings?

Of course, yes. He married a cousin of his mother

 

[bob012345]

Of course not! Is this a serious question?

It seemed you were suggesting that because you were answering my question of what Fermat may have thought.

 

[fresh_42]

What was Fermat's level of mathematical knowledge time?

Was it comparable to what Andrew Wiles knew?

I’ve read that Fermat would write to other mathematicians of his time, presenting a problem without providing his own solution. Hence, there are few known proofs done by him.

Additionally, mathematicians during his time were less rigorous than those of today's proof-oriented mathematics.

Fermat was a smart guy, but not at the level of the great French mathematicians, but we must not forget that we speak about the 17th century, not the 18th or 19th. Even Newton's Principa hadn't been published yet. I wouldn't compare him with Wiles. It is believed that he had a proof for n=4 and possibly n=3, and presumably thought it would generalize to any power. He shared the fate with Kafka and Galois: many of his thoughts were published by his executor, his son.

Yes, those letters asking someone else to solve a problem while already having a solution were a mean game at these times. However, I am not 100% sure whether Fermat was one of them.

 

[fresh_42]

Obviously Fermat thought he had a proof. Are there good candidate “proofs” people think may have been what he did and what are the typical flaws?

I have presented the proof for n=4 in the article, and n=3 can be done in a similar way. But it will be hard without using complex numbers.

 

[fresh_42]

It is unfortunate that you picked a near miss by one, since it has an obvious parity problem.

Near misses are extremely rare. I have found near misses of the order $\pm 10^{33}.$ Although they are still relatively close since the summands were of order $10^{39},$ I thought the parity problem would be less of a thing than the big gaps otherwise.

 

[fresh_42]

Did he have rings?

Rings, unique factorization in non-integer rings, elliptic curves etc. were all unknown. However, n=3 requires dealing with zeros of $x^2+x+1$ for which we would use complex numbers, but the people at the time were pretty resourceful and used their own environments surprisingly effectively.

 

[martinbn]

It seemed you were suggesting that because you were answering my question of what Fermat may have thought.

Yes, but I said rings because it was easier and shorter. Clearly you don't have to use this terminology.

 

[TensorCalculus]

you know, last year in school we were learning Pythag theorem, and I had just read Simon Singh's book on Fermat's last theorem. Seeing the similarity between pythagoras and what I had read about, I went up to my maths teacher after class and asked if he would be able to prove Fermat's last theorem since it wasn't in the book in detail and I was curious.
Poor guy didn't know what to say haha - it's only a few months later on that I eventually realise what a stupid question it was to ask... imagine being a maths teacher and one of your students just asks you to prove Fermat's last theorem on the spot, after class when they have another lesson to go to, and they're probably expecting like a 5 minute answer... whoops.

 

[jackjack2025]

you know, last year in school we were learning Pythag theorem, and I had just read Simon Singh's book on Fermat's last theorem. Seeing the similarity between pythagoras and what I had read about, I went up to my maths teacher after class and asked if he would be able to prove Fermat's last theorem since it wasn't in the book in detail and I was curious.
Poor guy didn't know what to say haha - it's only a few months later on that I eventually realise what a stupid question it was to ask... imagine being a maths teacher and one of your students just asks you to prove Fermat's last theorem on the spot, after class when they have another lesson to go to, and they're probably expecting like a 5 minute answer... whoops.

There are worse experiences as a professor than that :)

I have met Simon Singh. You may like the Code book, the Enigma stuff is interesting.

 

[TensorCalculus]

There are worse experiences as a professor than that :)

he teaches secondary school children (so 11-18 year olds) I can imagine there are probably worse things yes

I have met Simon Singh. You may like the Code book, the Enigma stuff is interesting.

whoa meeting Simon Singh is really cool! I have read the code book, personally I prefer his book on Fermat's last theorem but that one is really good too :)

 

[jackjack2025]

he teaches secondary school children (so 11-18 year olds) I can imagine there are probably worse things yes

whoa meeting Simon Singh is really cool! I have read the code book, personally I prefer his book on Fermat's last theorem but that one is really good too :)

What the teacher should have said is: Yes of course I can prove it, but the margin is too narrow to contain the proof (or some variation of that get out clause)

 

[martinbn]

you know, last year in school we were learning Pythag theorem, and I had just read Simon Singh's book on Fermat's last theorem. Seeing the similarity between pythagoras and what I had read about, I went up to my maths teacher after class and asked if he would be able to prove Fermat's last theorem since it wasn't in the book in detail and I was curious.
Poor guy didn't know what to say haha - it's only a few months later on that I eventually realise what a stupid question it was to ask... imagine being a maths teacher and one of your students just asks you to prove Fermat's last theorem on the spot, after class when they have another lesson to go to, and they're probably expecting like a 5 minute answer... whoops.

I don't see what the problem is.

 

[jackjack2025]

I don't see what the problem is.

digging out Andrew Wiles' proof and explaining it on the spot would be a bit of a problem, no?

 

[martinbn]

digging out Andrew Wiles' proof and explaining it on the spot would be a bit of a problem, no?

I meant, i don't see the problem if a student asks the teacher how to prove Fermat and the teacher diesn't know.

 

[nomadreid]

I meant, i don't see the problem if a student asks the teacher how to prove Fermat and the teacher diesn't know.

As a longtime mathematics secondary school teacher, I know the solution to this situation if one doesn't want to admit that one doesn't know: "Excellent question! That's your homework assignment for tomorrow."

 

[TensorCalculus]

@martinbn - I ended up searching up the proof a little while later and it's pretty long, can't imagine someone would have it memorised... or if they did be able to explain it to a 12 year old kid (who, sure, is a huge nerd, but also 12 so uhm yeah said kid's maths knowledge is really limited) in less than 5 minutes... I mean it's just a bit awkward I guess, and looking back on it was probably not the best question to ask haha

 

[fresh_42]

I once received a very unsatisfactory answer to a question I asked my math teacher, too. He basically said it was difficult and beyond my horizon. I don't think this was true. He could have said: I currently don't know, but I will look it up and tell you tomorrow, for example.

In my opinion, the best answer about FLT would have been: It uses a standard proof technique by transforming the problem into another, more general one that can then be solved. However, this more general problem dates back to a conjecture made in 1958 that itself requires higher mathematics to prove. The fact that FLT has been unsolved for 350 years and the other one for 35 years can be viewed as evidence of their complexity.

 

[TensorCalculus]

Yeah that's probably quite a good way of putting it - in the end he just said that it would take a while to explain so I should look it up and then ask him if I had any questions - spoiler alert I didn't have any questions because unsurprisingly I didn't understand a thing . Not as bad as just going "it's difficult" though I guess :)

 

[fresh_42]

Yeah that's probably quite a good way of putting it - in the end he just said that it would take a while to explain so I should look it up and then ask him if I had any questions - spoiler alert I didn't have any questions because unsurprisingly I didn't understand a thing . Not as bad as just going "it's difficult" though I guess :)

There is also the possibility of saying: Solve it for $n=2$ and use this result on $a^4+b^4=c^2$ to solve it for $n=4.$ Still a bit tricky, but doable for a 12-year-old. This risks frustration for the promise of the feeling of success. I'm not sure whether I would risk this in the role of a teacher. On the pro list is that the case $n=3$ then paves the way to think about complex numbers.

 

[TensorCalculus]

There is also the possibility of saying: Solve it for $n=2$ and use this result on $a^4+b^4=c^2$ to solve it for $n=4.$ Still a bit tricky, but doable for a 12-year-old. This risks frustration for the promise of the feeling of success. I'm not sure whether I would risk this in the role of a teacher. On the pro list is that the case $n=3$ then paves the way to think about complex numbers.

hmm, 1 year older and 1 year wiser, I might try that and see what I get it sounds fun :)
and try n = 3 too of course

 

[fresh_42]

hmm, 1 year older and 1 year wiser, I might try that and see what I get it sounds fun :)
and try n = 3 too of course

Ask for help / hints here before frustration takes over.