INCORRECT PROOF of Fermat's Last Theorem

before you bring a trampoline to this bowling tournament you should know that your cousin Archy Dyfunkle is the referee and dating your mother-in-law. There's a strict and I mean STRICT "NO TRAMPOLINES" allowed sign outside the combination Taco Bell-KFC-Bowling Alley. Okay folks sorry we're going to have to call off the butter chugging contest. if you watch this video till the end, a magic math moth may make mother moose milk made mischievously matching maroon Mariners merrily marching madly mooning magazine merchants.

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[mathfincoding]

"We most definitely cannot fit the proof of Fermat's Last Theorem on this chalkboard." I see what you did there.

[tyrjilvincef]

I was expecting the false contradiction to be more subtle but they pretty much just gave up and shat their pants.

[VideoFusco]

The author of the article was referring to Niven's theorem, according to which cos(theta) and theta (in radians) are never both rational except for theta=0. The author of the article has completely misrepresented this theorem because in its construction theta can be irrational.

[manucitomx]

The description was great.
Also, showing errors is very instructive.
Thank you, professor.

[TJStellmach]

Here I was keeping a lookout the whole time to try to pick where the proof falls apart, and the twist turns out to be that the problem with it is glaringly obvious!
(Not, as Prof. Penn says, that there couldn't be a reasonable proof based on the premise that some derived quantity is rational when it couldn't be, but "any value whatsoever of cosine" isn't that.)

[stratehorthy]

There is another issue with the shortening of triangle. For x, y, z to be sides of a triangle, they need to satisfy the triangle inequality. Now we know for sure that x, y, r are sides of a right triangle and we know that z < r, but if we assume wlog that x>=y, then it might very well be the case y+z<x. To fix this, we need to use the detail that z>x and z>y, and this tells us that z+x>y and z+y>x. Now we can use the fact that x+y>r>z to conclude that we can infact shorten it to be an appropriate triangle.

[alekseikhalin]

Hopefully, this comment will shed some light on the situation.

TL;DR: author actually believed the proof to be correct, but he wasn't a number theorist.

I decided to find the original article myself, which proved to be a challenge, since the link provided by the blogspot didn't lead anywhere. Fortunately, from the link I was able to identify the name of the newspaper, went to their archives, and searched via key words.

The name of the article loosely translates to "May the humanity rest now?". It goes into history of Fermat's Last Theorem, how Euler and Gauss were unable to solve it. It also briefly mentions one russian mathematician who presumably went insane trying to solve it (I heard nothing about such a story). The article is rather poorly written and unpleasant to read, especially where there are any mathematical formulas involved (they are written 'as is', for example, FLT is "xn + yn = zn").

The article then tells the story of Alexander Ilin, author of the proof. Funnily enought, he studied at the university in my hometown. It is stated that Ilin worked on a space program, and then for no reason at all the article mentions how Alexander was taken to court for money laundering and how he defended himself. After that, he dedicated himself to proving FLT.

Finally, the article states that Ilin came up with a proof and showed it to his collegues, who all agreed that they can't locate errors in it (while stating that they are not number theory specialists and are unqualified to say for certain). The article gives the proof, which is hard to understand because of how math is written, but it is at least almost identical to the one shown in the video.

So, long story short, Alexander actually believed he had found the proof, but he was never a theoretical mathematician, let alone a number theory specialist, so the error is somewhat understable.

[idolgin]

In Soviet Russia Fermat's Last Theorem proves you!

[kpaasial]

I wonder if Fermat's famous margin note was referring to a proof similar to this but he probably figured out his error quickly and didn't correct his margin note for whatever reason.

[scp]

One of my Professors once told me, that he received at least 10 (wrong) proofs of FLT (mostly from non-mathematicians ;-) ) a week.

[randomjin]

Tbh, this could be made a whole video series. All too much and too often we forget to look at past failures. But failure often teaches way more than success. On this example: I've definitely learned something from this whereas even if I tried to digest the correct proof of the theorem, I'd likely fail on page 1 and would need to do a two years preparation course to even start with it.

[Noam_.Menashe]

I'm going to need a source for cos(pi/6)=1/2.

[tobyfitzpatrick]

Fermat decided not providing a proof was a good place to stop.

[giovanicampos]

During my undergraduate I was solving a problem like "If x, y and z are sides of a right triangle such that z is the hypotenuse, show that z³ > x³ + y³".
From that I generalized for all n≥3 natural, we have zⁿ > xⁿ + yⁿ, where x, y and z are a pythagorean triple.
I mentioned this to my professor claiming that I proved Fermat's last theorem for a tiny subset of natural numbers =D

[mathboy]

This reminds me of when I proved Fermat's Last Theorem. I was using inequalities, trying to exploit Roth's Theorem by using the Taylor polynomials and remainders for y = (1-x^n)^(1/n). It taught me the danger of "working backwards" when dealing with inequalities. I found the error after hunting through it for at least an hour (assuming you're an idiot is always a safer bet than assuming you're brilliant). So fortunately I never showed anyone my "proof".
(The essence of my error worked like this: If you only have that x > 10, and your desire is to prove that x > 20, then to see what logical steps it would take to get from x > 10 to x > 20 (using a few other things you know about x), start at the finish line by assuming x > 20 and then work backwards to see if you can't find the path to x > 10. Sure enough, if you start with x > 20, you can indeed connect all the dots back to x > 10, and so you have your proof!)

[alastairbateman]

It is acknowledged at 4:09 that every power is an integer squared multiplied by the integer raise to the power minus 2. Basic laws of indices. Therefore a Fermat triple is impossible for every power above 2 due to the Distributive Law of Multiplication. Q.E.D.. So yes, a complete and absolute proof does fit on the chalk board with space to spare.

[andrewdsotomayor]

Imo the best false proof of FLT is the one by Lamé that falsely assumes unique factorization, and ultimately leads to proving FLT in the particular case when n is equal to a Sophie Germain prime

[almafater]

2:36 "and that's because we definitely cannot fit the proof of Fermat's Last Theorem on this chalkboard".

So you're saying that... you have a truly marvelous proof of this, which this chalkboard is too narrow to contain? 😅

[kazebaret]

Very interesting video about how mathematical proofs are elaborated. Small correction: I believe that cos(pi/3) = 0.5, not cos (pi/6) :)

[girla]

For a moment there I expected we were going to try and (wrongly) prove that theta is greater than pi/2, thus the contradiction. Kind of a let down that the author went with what he did.