# Did Pierre De Fermat *really* have a proof for Fermat's Last Theorem?

In 1637, Pierre De Fermat famously wrote in the margin of a copy of Arithmetica that he had proof that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n greater than two.

Given that Andrew Wiles' proof made use of a lot of new mathematics; for example, Ribet's 1986 epsilon conjecture proof and the Taniyama–Shimura conjecture, is it generally still believed that Fermat *did* have a general proof to this conjecture?
i sure did.
Well.. It seems like it. If you had read about his life, he was an advocate who did maths as a hobby. Every night, he does maths before goes to sleep and then, he came up with this theorem. On the same page he wrote this equation, on the top left corner he wrote "my wife is calling me. I know how to solve it. But, I don't have time!". Who knew? Maybe he was faking it. However, I think he had it.
Dammit fermat! Worst cliffhanger by far!
Original post by Defensive Gnome
In 1637, Pierre De Fermat famously wrote in the margin of a copy of Arithmetica that he had proof that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n greater than two.

Given that Andrew Wiles' proof made use of a lot of new mathematics; for example, Ribet's 1986 epsilon conjecture proof and the Taniyama–Shimura conjecture, is it generally still believed that Fermat *did* have a general proof to this conjecture?

No. He didn't. I would subscribe to the belief that he had something similar to Lame's claimed proof, as it is plausible that he could have come up with this, but the reasons it is wrong were very subtle by the standards of the time so he could have gone decades without realising it was wrong.
Apparently this explains it, but it sounds like BS to me.

http://www.fermatproof.com

Fermat had a proof in the case of fourth powers only, the rest weren't discovered if they were known at the time and I doubt he had the mathematical knowledge to get something anything like Wiles' proof, so I have to say no.

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Doubt it.

Simon Singh has a great book on the subject.
Original post by majmuh24
Apparently this explains it, but it sounds like BS to me.

http://www.fermatproof.com

Fermat had a proof in the case of fourth powers only, the rest weren't discovered if they were known at the time and I doubt he had the mathematical knowledge to get something anything like Wiles' proof, so I have to say no.

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That site made me Why would someone waste their time like that?
Original post by LightBlueSoldier
That site made me Why would someone waste their time like that?

I have my own amazing proof of this statement, but the reply box on TSR is too small to contain it.

I have absolutely no idea, seems like BS on a whole new scale.

It has been proven that any statement can be proven using basic laws of arithmetic and a weak form of induction, but I doubt that would be small enough to fit into any margin.

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(edited 10 years ago)
Original post by Defensive Gnome
In 1637, Pierre De Fermat famously wrote in the margin of a copy of Arithmetica that he had proof that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n greater than two.

Given that Andrew Wiles' proof made use of a lot of new mathematics; for example, Ribet's 1986 epsilon conjecture proof and the Taniyama–Shimura conjecture, is it generally still believed that Fermat *did* have a general proof to this conjecture?
Just to add: it is entirely possible (indeed many professional mathemticians consider it very likely) that there is an 'elementary' proof of FLT which uses even less than what was known in Fermat's time. However, if such a proof exists, it would be 'very long'.
Original post by Defensive Gnome
In 1637, Pierre De Fermat famously wrote in the margin of a copy of Arithmetica that he had proof that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n greater than two.

Given that Andrew Wiles' proof made use of a lot of new mathematics; for example, Ribet's 1986 epsilon conjecture proof and the Taniyama–Shimura conjecture, is it generally still believed that Fermat *did* have a general proof to this conjecture?

It is thought that Fermat assumed that all rings of the form $\mathbb{Z} \lbrack \sqrt{-n} \rbrack$ are unique factorisation domains - I'm told that it's not hard to prove FLT under that assumption. (The assumption is wrong - witness $\mathbb{Z} \lbrack \sqrt{-5} \rbrack$, in which $6 = 2 \times 3 = (1+\sqrt{-5}) \times (1-\sqrt{-5})$.)
(edited 10 years ago)
Very much doubt it. The mathematics needed to proof the theorem is much more beyond his abilities imo. The most probable scenario is he thought he had the proof, but it was actually wrong.
Original post by Smaug123
It is thought that Fermat assumed that all rings of the form $\mathbb{Z} \lbrack \sqrt{-n} \rbrack$ are unique factorisation domains - I'm told that it's not hard to prove FLT under that assumption. (The assumption is wrong - witness $\mathbb{Z} \lbrack \sqrt{-5} \rbrack$, in which $6 = 2 \times 3 = (1+\sqrt{-5}) \times (1-\sqrt{-5})$.)

This is the most common theory. (This was the Lame proof I mentioned above that Kummer corrected)

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Original post by Defensive Gnome
In 1637, Pierre De Fermat famously wrote in the margin of a copy of Arithmetica that he had proof that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n greater than two.
Given that Andrew Wiles' proof made use of a lot of new mathematics; for example, Ribet's 1986 epsilon conjecture proof and the Taniyama–Shimura conjecture, is it generally still believed that Fermat *did* have a general proof to this conjecture?

Dear Colleagues!

I am sending you my article as my friends have suggested.
I will now present my friend's speech from the IAEA:

Fermat's Theorem - A Proof by Fermat Himself
(c) Yurkin Pavel, IAEA

The Russian nuclear physicist Grigoriy Leonidovich Dedenko has reconstructed the original reasoning of Pierre Fermat, which led Fermat to conclude that the sum of two identical natural powers of rational numbers, raised to an exponent greater than two, is not representable. This is known as Fermat's Last Theorem.

As you may know, in 1637, Fermat wrote a note in the margins of his copy of Diophantus's "Arithmetic" stating his discovery and adding, "I have discovered a truly marvelous proof, but this margin is too narrow to contain it."

According to G.L. Dedenko, Fermat analyzed power differences using a method that was novel at the time: decomposing these differences into a sum of pairwise products, later known as the "Newton binomial". Fermat discovered that the coefficients in this expansion satisfied simple conditions equivalent to a logarithmic equation (a concept still developing in the mid-17th century) for the degree of the sum (or difference). This equation has only two solutions: the numbers one and two.

Thus, the margins of the book were indeed too narrow to contain the complete proof. Fermat's proof would have required the introduction and development of new concepts, such as expansion with combinatorial coefficients (Newton's binomial) and logarithms. It remains unclear whether Fermat ever wrote down his detailed reasoning, and if so, whether this record survives in an unexpected archive. Historians of natural history are encouraged to search anew.

Please see the final version No. 25
Fermat's Theorem - A Proof by Fermat Himself

Sincerely,
Ph.D, Grigoriy Dedenko
15.jun.2024