NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) This video is about Fermat's last theorem and Euler's conjecture, a vast but not very well-known generalisation of this super theorem. Featuring guest appearances by Homer Simpson and the legendary supercomputer CDC6600. The video splits into a fairly easygoing first part and a hardcore second part which is dedicated to presenting my take on the simplest proof of the simplest case of Fermat's last theorem: A^4 +B^4=C^4 has no solution in positive integers A, B, C. The proof in question is taken from the book Lectures on elliptic curves by J.W.S. Cassels (pages 55 and 56). Here is a scan of the relevant bits: http://www.qedcat.com/misc/cassels_proof.pdf This writeup of the proof actually contains a few little typos, can you find them? In the video I attribute the proof to John Cassels the author of this book because I've never seen it anywhere else. It's certainly not Fermat's proof as one may be led to believe reading Cassel's writeup of this proof. The Wiki page on Euler's conjecture contains a good summary of the known results and a good list of references: https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture There is one aspect of this conjecture that I did not go into in the video. The conjecture says that for n greater than 2 at least n nth positive integer powers are necessary to make another nth integer power. On the other hand, it is not known whether for every such n there is an example of n nth powers summing to another nth power. In fact, even my example of a sum of five 5th powers summing to another 5th power in the video was not known to Euler. Anyway, the wiki page also has a summary of what's known in this respect. Today's t-shirt should be easy to find, just google what it says on the t-shirt. Thank you very much to Danil for his continuing Russian translation support, Marty for his very thorough nitpicking of the script and all this help with getting the explanations just right and Michael for his help with filming and editing. Enjoy! Typo: (Someone who's really paying attention :) Great video as usual (even though I already knew the proof). There's a small mistake at 17:48, on line 5 it should be (u^2-Y)(u^2+Y)=4v^2 instead of (u^2-Y)(u^2-Y)=4v^2. (M) Yep, luckily not where I actually do the proof. Actually a great one to pinpoint who is really paying close attention to detail :)