Fermat's Last Theorem and the Mysteries That Remain

In this episode, first I'll explain the historical tale of Fermat's Last Theorem, and then show some other mysterious questions, including a conjecture Euler got wrong and the unsolved $1,000,000 "Beal conjecture"!
[Disclaimer: Do not copy any uses of fire that you may see in this episode].
Here's my previous episode about Pythagorean triples: • Pythagorean Triples (a...
Make sure you're also tuned in to my livestreams and other bonus content on my ‪@Domotro‬ channel!
This was filmed by Carlo Trappenberg.
Special thanks to Evan Clark and to all of my Patreon supporters:
Max, George Carozzi, Peter Offut, Tybie Fitzhugh, Henry Spencer, Mitch Harding, YbabFlow, Joseph Rissler, Plenty W, Quinn Moyer, Julius 420, Philip Rogers, Ilmori Fajt, Brandon, August Taub, Ira Sanborn, Matthew Chudleigh, Cornelis Van Der Bent, Craig Butz, Mark S, Thorbjorn M H, Mathias Ermatinger, Edward Clarke, and Christopher Masto, Joshua S, Joost Doesberg, Adam, Chris Reisenbichler, Stan Seibert, Izeck, Beugul, OmegaRogue, Florian, William Hawkes, Michael Friemann, Claudio Fanelli, The Green Way, Julian Zassenhaus, Bailey Douglass, Jan Bosenberg, Brooks Boutwell, David Irvine, que and George Sharabidze!
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In case people search any of these terms, some of the topics discussed in this episode are: Pythagorean triples, Pierre de Fermat, Fermat's Last Theorem, Euler's Sum of Powers Conjecture and similar questions in number theory, the Beal conjecture, Andrew Wiles, and much more!
DISCLAIMER: Do not copy any uses of fire, sharp items, or other dangerous tools or activities you may see in this series. These videos are for educational (and entertainment) purposes.

[@ComboClass]

Thanks for watching! Leave a comment with your favorite unsolved question. And if you haven't seen yet, here's my previous episode about the related topic of Pythagorean triples/quadruples: kzbin.info/www/bejne/raWyf6CojraUh7c

[@roberto4898]

You are one of my favs in KZbin. Keep it goin'!

[@socalacura1338]

The Riemann Hypothesis and P vs NP have got to be my two favorite unsolved problems in math/computation

[@DarthMauldinOfficial]

I love how you talk about numbers. Like how 3,4,5 is the “coolest” pythagorean triple, but it has an “underrated older brother.” I love your enthusiasm, and the utter chaos each new video contains.

[@pedramhashemi5019]

I have a truly marvelous solution to Riemann hypothesis, which this comment box is to short to contain

[@ComboClass]

Yeah I have a hilarious response to that comment that would make everyone crack up but similarly this reply box is too narrow to write down the hilarious reply

[@pedramhashemi5019]

​@@ComboClassyou literally cracked me up on 5:18, so much I had to pause the video, come down and write that reply, and then go back to emd the video 😂😂

[@MarkAhlquist]

Plus you'd have to use emojis.

[@chem7553]

K p li9900 lol o0plp9😊

[@okarakoo]

This guy is a natural born lecturer, with his long takes without "uum", "ehm" or any other filler and very clear diction. Much envy -)

[@WordToMomsYo]

Just wanted to pop in with a quick comment.. I LOVE the editing and adding of additional footage that I see here in this video.. it adds humor and just makes the whole video that much more visually appealing. Keep up the good work!

[@publiconions6313]

Wonderful video!!.. my son (10yrs and already very much into math) and i enjoy watching together. You've really found a unique voice which is a rare thing.. and Carlo as always does a great job too.

[@soninhodev7851]

great video as always, i love the direction the channel is going. its so fun to see all the mathetical theory yet to be discovered! =)

[@nitehawk86]

Love the Ferrmat death renactment. I can imagine a scenario he died just after right after writing the note, such as in Monty Python, "Perhaps he was dictating?"

[@MarloTheBlueberry]

bro died for us and came back to life just to educate us.. We need more teachers like this..

[@abhishekak9619]

everyone says they want more like this but nobody wants to be like them. /s

[@quarkonium3795]

Small correction: Wiles did not prove the Taniyama-Shimura conjecture entirely (the one relating elliptic curves to modular forms). Turns out the Fermat's Last Theorem is only dependent on a weak form of the Taniyama Shimura conjecture that deals with semistable elliptic curves. The actual TS conjecture was proven by Conrad, Diamond, Taylor, and Breuil in 2001; building off the work done by Wiles and Taylor to prove Fermat. It's now known as the Modularity theorem and it is in my opinion the most important mathematical theorem proven within the last hundred years

[@HampterLover985]

All that wood around the fire is sweating 👀💧

[@aguyontheinternet8436]

I KNOW the table is waaaaay too close, same with the clock and the _side of the building._ I hope there was a fire extinguisher on hand, cause things coulda gone real south real quick

[@MagruderSpoots]

And the neighbors.

[@Matt-mq6ws]

You actually only need one of any given power to add up to a number of that power. For example: 5 cubed is equal to 5 cubed

[@mistafizz5195]

I love the props

[@TalsBadKidney]

You got to dig it to dig it

[@hkayakh]

What about cases where the exponent is less than 1? Or less than 0? Or in between 1 and 2? Do those work, not work, or have no proof?

[@landsgevaer]

If you allow real-valued exponents, there are infinitely many examples. For example, since 4¹ + 5¹ > 7¹ and 4² + 5² < 7² there must be an intermediate exponent for which equality holds (since exponentiation with positive base is a continuous function). For exponent -1 there are infinitely many solutions as well, like 1/(n+1) + 1/(n²+n) = 1/n Would have to give the general case of negative integers more thought.

[@deinauge7894]

but for almost all real numbers there is no solution. Because there are only countably many triplets of whole numbers a,b,c and therefore countably many solutions to a^x+b^x=c^x

[@bellanova1635]

Amazing channel- modern day Bill Nye vibes. Are you on Twitter? Brilliant minds are hard to ignore and you just need to reach that one person that can skyrocket your channel and your future. Wishing you health and happiness in all that you do ↔️&♾

[@ComboClass]

Thanks! I’m not on Twitter. There are some links in the description for the other places I am online, like my bonus channel, discord, and etc.

[@stickmcskunky4345]

It gets the hose

[@hkayakh]

I gotta say, you’re one of the more surreal mathematician. Lots of things break in your math videos. I bet there’s some ‘’’broken math theorum’’’ you’d cover if it existed

[@iamdigory]

Godel's incompleteness theorem?

[@calebellis7295]

I’m still wondering how many credit hours I need for my degree from combo class

[@otonanoC]

I'm a graduate student. ANd I never knew that 3^3 + 4^3 + 5^3 = 6^3 until I saw this video.

[@wrc1210]

When he showed 3^3 + 4^3 + 5^3 = 6^3 did anybody else go to your calculator and try 3^4 + 4^4 + 5^4 + 6^4 to see if it equals 7^4?

[@basqye9]

Fermat-- maybe the greatest troll in human history.

[@Kopiovastaava]

Videos of Combo Class are really something else. Most of the time I feel whole, maybe even a bit rational if I'm not too negative. But sometimes when I can't construct my thoughts I get irrational and start wondering something imaginary. All the distractions, weird camera angles and random stuff in these videos feel surreal yet it all somehow computes.

[@BuleriaChk]

Proof of Fermat's Last Theorem for Village Idiots (works for the case of n=2 as well) To show: c^n a^n + b^n for all natural numbers, a,b,c,n, n >1 c = a + b c^n = (a + b)^n = [a^n + b^n] + f(a,b,n) Binomial Expansion c^n = [a^n + b^n] iff f(a,b,n) = 0 f(a,b,n) 0 c^n [a^n + b^n] QED n=2 "rectangular coordinates" c^2 = a^2 + b^2 + 2ab Note that 2ab = 4[(1/2)ab] represents the areas of four right triangles) "radial coordinates" Lete p:= pi, n= 2 multiply by pi pc^2 = pa^2 + pb^2 + p2ab Note that pc^2, pa^2, and pb^2 represent areas of circles, wile p2ab = a(2pb) is the product of a radius (a) and a circumference (2pb). This proof also works for multi-nomial functions. Note: every number is prime relative to its own base: a = a(a/a) = a(1_a) a + a = 2a (Godbach's Conjecture (now Theorem.... :) (Wiles' proof) used modular functions defined on the upper half of the complex plane. Trying to equate the two models is trying to square the circle. c = a + ib c* - a - ib cc* = a^2 + b^2 #^2 But #^2 = [cc*] +[2ab] = [a^2 + b^2] + [2ab] so complex numbers are irrelevant. Note: there are no positive numbers: - c = a-b, b>a iff b-c = a, a + 0 = a, a-a=0, a+a =2a Every number is prime relative to its own base: n = n(n/n), n + n = 2n (Goldbach) 1^2 1 (Russell's Paradox) In particular the group operation of multiplication requires the existence of both elements as a precondition, meaning there is no such multiplication as a group operation) (Clifford Algebras are much ado about nothing) Remember, you read it here first) There is much more to this story, but I don't have the spacetime to write it here.

[@Edmonddantes123]

KZbin’s most criminally underrated channel!

[@g_vost]

genuinely it took me like a solid 15 seconds to realize that youre standing next to an OPEN FIRE

[@finnboltz]

Now, if you're so interested in fun stories from the history of maths, why haven't you made a video on the history of the cubic and quartic formulae yet?

[@donwald3436]

I mean if he really did have a proof he could slip an extra piece of paper between the pages? lol

[@hkayakh]

3:36 cat moment Oh also if you ever publish a math paper, will you include your cat(s) as a co-writer?

[@yasin_karaaslan]

I laughed pretty hard when you showed the shortest paper of all time haha

[@MarloTheBlueberry]

Combo, you should he lucky! I am dedicated enough to watch this video at midnight. . . Ok so listen here bub, where's my -2 Degree f or Combo Class?

[@MarloTheBlueberry]

Nah mate, I'm serious, WHERES MY DEGREE

[@ComboClass]

@@MarloTheBlueberry Combo Class doesn't care about degrees. However, you are definitely gaining levels of combo lord experience :)

[@MarloTheBlueberry]

yay combo lord experience :D

[@Eevi-oi]

I want to eat mushrooms with this guy

[@funnywarnerbox300]

Combo Class is going off lately, working overtime

[@angusmacdonald4860]

Hey bro, can you do a video on the Zeroth root

[@benjaminsowers1550]

bro has the lordvessel

[@ronald3836]

Someone please save that Rubik's cube from melting!

[@ComboClass]

Yeah don’t want to turn it into a rubix sphere by accident haha

[@JurassicJenkins]

Humm… ❤

[@RibusPQR]

What are your cats' names?

[@ComboClass]

Dandelion, Sage, and Sassafras

[@oriongurtner7293]

YAY YOU DID THE CUBIC ONES!! I love those, they don’t prove Fermat’s Last, but they do back it up

[@MrCheeze]

I think the conjecture at 12:00 is probably false. Not only because it's similar to Euler's false conjecture, but also because there's just SO MANY degrees of freedom to choose candidate numbers for large n.

[@landsgevaer]

Many degrees of freedom? The number of integers is the same as the number of n-tuples of integers, so in that sense you actually have the same "degree of freedom" (one aleph-zero).

[@aethrya]

My man Domotro posted a new video! Dope.

[@drakewilson6913]

Love this guy

[@lugyd1xdone195]

A, b, c cant equal zero. Just a bit that annoyed me.

[@ComboClass]

Yeah I didn’t mention the 0th power since 0 is not a “positive integer” which the exponents are usually restricted to, but yeah trying 0th power would make 1+1=1 which obviously won’t work haha

[@lugyd1xdone195]

@@ComboClass no no no, thats also true, but I meant the 0^n + 0^n = 0^n as in a b c for all integer n. If any of the three is zero the equation is possible. Or is it that a cant equal b cant equal c?

[@ComboClass]

@@lugyd1xdone195 Ah. The base numbers there are still required to be positive integers in this case, and 0 is not "positive"

[@lugyd1xdone195]

@@ComboClass ahh, thanks. Now I need to know if it still holds for negative whole numbers. For every even integer n it has to hold, but what about the odd ones? This is what confused me in the video because as far as I remember you mention the base numbers have to be whole and n has to be an integer.

[@tom.prince]

One thing I think I have read is that later in life Fermat himself wrote a proof for one particular case of n > 2 (I think 3 or 4), which suggests that he may have later realized that his claimed marginal proof was incorrect.

[@ComboClass]

It also could have gone the other way around, where his discovery of a particular exponent case came first and then his margin note was based on him believing he had figured out how to extend that technique to a general case. But yeah the historical details are not fully known

[@ronald3836]

Fermat did n=4 (which is relatively easy). Later Euler did n=3.

[@BillGreenAZ]

It seems to me I saw a much simpler proof of Fermat's Last Theorem. A guy used square blocks and piled them up to show how units were used to add up to powers. I wish I could find that again.

[@ComboClass]

Be warned, like I said in the episode, there are literally thousands of false proofs for it out there. And there isn’t an accurate proof (at least not yet) that isn’t many pages

[@BillGreenAZ]

@@ComboClass Agreed. I am well aware of all those erroneous "proofs". It just seemed that this way of "proving" the theorem seemed to be so simple and so logical that I could see how this would fit Fermat's original statement.

[@renerpho]

As of 2023, Wiles' proof essentially remains the only one. No alternative approach to using the "Modularity Theorem" has yet been found. Not only that, but nobody has even found an alternative proof for the Modularity Theorem.

[@renerpho]

And it's not for lack of trying! Finding a second (simpler) proof for the Modularity Theorem would be a big deal.

[@BillGreenAZ]

@@renerpho I agree that it's very likely that the "proof" I came across with the blocks was not a valid proof. However, there was some elegance in its simplicity.

[@paulytools]

yeehaw

[@stickmcskunky4345]

This one is absolute fire

[@delwoodbarker]

I can smell the smoke even from here