Ramanujan, 1729 and Fermat's Last Theorem 

Video length - 16:47. You couldn't stretch that out another 42 seconds? :)

Damn, Ramanujan is even more genius than I thought! I love how he was just dancing with numbers and came up with infinitely near misses for Fermat's last theorem like nothing. A genius playing with another genius's work.

"... before you know it you get a negative twelfth out the other side and everyone gets very emotional." That comment literally made me laugh out loud. Wonderful presentation. Thanks. Jon PS: And then I got to the end and laughed out loud yet again. :)

I felt my heart crack when he said that ramunjan was so close to solving Fermat's last theorum

I guess 9^3 + 10^3 is a Parker Square of a counter argument to fermats last theorm

Just imagine if Ramanujan had proper education and lived long life.

8:53 So Ramanujan was looking for a bit of a Parker Square of a solution.

Infinite series, Number Partition Theory, mock theta functions there are so many work of him..many of us don't know that his number theory is used to generate credit/debit card and currency numbers...That's Ramanujan .

Imagine if he was around now, imagine him doing a numberphile video.

You get a negative twelfth on the other side and everyone gets very emotional. Well Ramanujan did get negative twelfth and even wrote about it to Hardy.

I always get a bit hyped when I see people I like on youtube, encouraging us to watch other people I like on youtube. I watched James' videos just before this one, and they were excellent, and I have always liked Simon's Numberphile video, and this is the first time I've seen any other youtuber recommend Mathologer. I really Dig Mathologer/Burkard's channel, and I especially liked his -1/12 vid. I'm glad someone finally mentioned him, and not surprising it was you, cause Matt is also a boss. Thanks for that, and another captivating video too, of course.

I love your content. It's entertaining and you don't water down the maths involved in your videos. Keep up the fantastic work.

12:26 "infintie series"... the parker square of typing.

It was a great film!! Lots of emotion and many great actors helped life his story to the screen

Mathologer's -1/12 video is pretty great. I like the way he goes into detail about the ways people have played with diverging infinite series.

Thanks for the lecture scan, Matt. I look forward to giving it a read at some point when I'd really be better off working.

9:00 Also thanks to Simon Singh, we know those near misses are used in The Simpsons.

I enjoyed the movie. Thats y i wanna know more about this person and his formulas.

Best quote: "You start putting numbers in, Before you know it you get a -1/12 out the side and everyone get very emotional", LOL,!

"Not by a physicist" Shots fired

Thank you so much for scanning the first chapter of the book!

Ramanujan did a Parker's Square proof of Fermat last theorem...

I frickin love the oozing enthusiasm! Thanks Matt. Btw, not a mathematician and the movie was grand.

65601^3+67402^3=83802^3 is quite the parker square of a counter example!

This was a beautiful video, i appreciate it! I'm a big fan of Ramanujan

It's a bit depressing to think about the fact that despite of how hard you work, you will never be even close to the intellect of some people.

Brilliant ! Simply love the way you explain it Sir !

Thanks for the update and the valuable video, I appreciate you

Interesting! I've come across generating functions when studying probability. Specifically, moment generating functions for the moments of a particular probability distribution.

I Like how mathematics is timeless more or less; watching this in 2020. Could have been in 2016 or 1729 as well. Kudos Parker.

Man, this video went all over the place. What a wild ride

You are the best I learn from your videos as opposed to other mathematicians on RU-vid ranting and wasting my time

He would have solved fermat's last easily if he hadn't died. God knows why he dies so young, it's insane.

1. excellent movie review 2. nice build on 1729 3. love the eyebrows ; p 4. please keep posting : )

Very interesting video by you. Only video of such type on RU-vid. Praise your researches on The Legendary Ramanujan.

This guy manages to glue one to his lecture on such an uninteresting Subject- mathematics. Thank you very much!

Sad to see that he isn't given enough credit in his own fatherland,India. Nobody teaches about the great Mathematician in schools & colleges. Hollywood came up with his biography but our own Film Industry isn't even thinking of it. Anyways, it's too much to expect the biography of some great Mathematician when more than half of the Hindi Film Industry is below par educated to understand the beauty and significance of Mathematics.

I found them. I did it wrong the first time which resulted in my changing over 200k values in a spreadsheet at once which took a while, and even longer to render the colors, but when I made my grid only 159x159 like I was supposed to, was way easier. 158^4+59^4=134^4+133^4=635,318,657

16:12 I love Matt's tiny laugh at the comparison between himself and Ramanujan

My university's library has a copy of this book Should probably pick it up and give it a read

11:37 good one!

That was a fantastically whirlwind trip tying together some seemingly remote threads! And it's especially poignant to realize that Ramanujan might well have proven FLT decades before it was actually done. I particularly liked the "near-miss" treatment of FLT solution attempts, because that's the way I've always looked at Pell's equation - no (non-zero) square can equal n times another, if n is a (positive) non-square (e.g., Euclid's proof that √2 is irrational); - but you can always find numbers that "miss by 1" - - and that is Pell's equation! p² = nq² + 1 . . as well as what might be called, the associated Pell's equation p² = nq² - 1 which has solutions for infinitely many n's, but none for infinitely many other n's.

For those that are interested, the book "generatingfunctionology" (yes that is the name) by Wilf goes into some of the interesting theory and somewhat incredible applications behind generating functions. It is also available as a PDF (legally) online. This type of thing is heavily used in analytic number theory and it is fascinating how many interesting properties can be deduced using such a simple idea.

Very good teaching. I recommend you make more video, you sure know how to explain. Thankyou so much. Learned today from a good teaching skill.

For those looking for the book: Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work (by G. H. Hardy) ISBN-10: 0821820230 ISBN-13: 978-0821820230

11:34 hahahaha you should do your own video on -1/12

4:12 for those wondering, this is the solution for 4th powers: (spoiler warning) 59^4 + 158^4 = 133^4 + 134^4

I usually use Numberphile videos to go to sleep and once it gets confusing I doze off while learning unconsciously. Matt Parker is just so fun and engaging that I just have to watch with full attention. Could you please please PLEASE explain Ricci flow and the Poincaré conjecture? I'm sure you of all people could describe it in a way that makes sense.

Dude ramanujan had everything; he was as genius, incredibly good looking

we are lucky that hardy did not dismiss ramanujan and gave him a chance.. else the world would have not known about Ramanujan the genius..

Matt, you and Ramanujan really are quite similar. His search for near misses regarding Fermat's Last Theorem reminds me very much of the Parker Square. But on a serious note, this is a great video and I'm going to check out all of the information you've linked us in the description now.

Yay! I am excited about this film & amazing mathematics :)

I got to see the film at last year's TIFF and got to sit beside George Andrews, the "researcher" who found the Lost Notebook, he was great, and the director Q&A after was fascinating! Also, read ALL of Simon Singh's books, they are excellent!

I went to visit him while he was lying ill at the hospital. I had come in taxi cab number 14 and remarked that it was a rather dull number. "No" he replied, "it is a very interesting number. It's the smallest number expressible as the product of 7 and 2 in two different ways." -- From M.O.

i love that Matt couldn't be bothered to fix the "infintie" blooper.

Mostly they are supposed to do with partitions. Partitions are a kind of things that can be used to see space and how they align. Say you take a cubicle of 9×10 the 12×1 cubicles can be created. By rearrangement of boxes. Easy to do volume calculations of products rearrangement.

11:31 "... and everyone gets very emotional". Gee, this sums up youtube comments about infinite sums so very well.

You could say it's a Parker Square solution to Fermat's Last Theorem.

love the Casual shots at sixty symbols physicists at the end :')

Very engaging. Just dipping my toe into maths via Machine Learning. Love this and Numberphile.

Thank you, Excel, for being a giant table with conditional formatting that allowed me to answer Matt's puzzle about fourth powers.

Thank you Matt for another great video and an interesting challenge problem. I thought that my process might be interesting. After one mistaken attempt, I was able to solve A^4 + B^4 = 635,318,657 in three tries. 1) For A^3 + B^3 = 1729, the sum is just one more that a cube. I hoped that one A would be a low single-digit. Knowing that our nine-digit target begins with 63 (almost 64), and remembering that the leading digits of powers of 2 are themselves close approximations of powers of 2 that are 1/1024 as large, I thought that one B could be close to 2 times a power of 10. Wrong! 635,318,657 ~ 2^6 x 10^7, the fourth root of which will NOT yield my hoped-for result. 2) What if A and B were approximately equal? Then A and B might be close to (635,318,657 / 2) ^ 1/4 ~ 133.5. Choose the integer part for A. Solution 1: A = 133 and B = 134! 3) What if the difference between A and B was the greatest? The fourth root of 635,318,657 is about 158.8; again choose A by selecting the integer part. Solution 2: A = 158 and B = 59! Luckily, my method led me to the solutions. This "least and greatest difference - integer part" strategy also works with 1729. Would it also work for the lowest number that is the sum of two squares in two different ways? For 65, the "greatest" portion yields A = 8, which is approximately the square root of 65 ~ 8.06, but "least"-wise would give us a guess of A = 5, as (65/2)^1/2 ~ 5.7. The second solution is instead A = 4, B = 7. Often times, however, for A^2 + B^2 = N, the "least difference" method, A ~ (N/2)^1/2, will give you a good first approximation, with the solution being near by. Keep up the good work!

You are awesome matt

12:10 especially nice since x^2 - x = 1 or x^2 - 1 = x is solved with 0,5+(1+(0,5)^2)^0,5 and i wrote it that way because it fits with the following term's logic

At school I sucked at mathematics ... somehow, 7 years later I am here looking up for mathematics videos ... can't understand much tho, but still kinda love watching it.

"No, it's a very interesting number, because when I see it on my clock, I know there's only one minute to go before going-home time."

thanks for the scan parker

I CANT CONTROLE MY TEARS HO WHAT AM SAY WHEN SEE RAMANUJAN'S STORY , YOU ARE GREAT HARDY SIR, YOU ARE GREAT

Dear Matt, I have found the PDF of the book here: eciencia.urjc.es/handle/10115/1436

fantastic video! very interesting, and this GH Hardy wrote very eloquently. he sounds easier to read than some modern writers of maths papers

You a my favourite mathematician

Ok now you need to do a video explaining K3 curves and algebraic geometry in general. Or where do you even start to learn that stuff?

It appears that Ramanugan might have cracked Fermat theorem had he lived another 20 years. That was an incredibly difficult proof and we should remember that Wiles needed help in the end. I think Galois theory should be introduced in high school, after all Galois had very little education, his name is on the first page of Wiles’ proof of Fermat

Wish India's school teachers at least recognised Ramanujan's talent beyond his Indianness.

Thanks for talking about the generating functions Matt. The way people usually tell the 1729 story, like he invented it out of thin air, seems misleading. I also believe this fact about 1729 was recorded in the notebooks years before the Taxi-Cab incident. I forget the exact page, but I think its on a page that also contains a geometrical problem (This narrows it down a lot since Ramanujan did so little geometry in his notebooks).

Ramanujan was a human calculator just like Scott Flansburg and Shakuntala Devi are. They can visualize numbers and see patterns in them. It's a gifted mind.

you should do a collab with cinemasins about it. you talk about the math while they talk about the cinematography.

will you do a separate video on generating functions please?

Thanks for posting..♥

Regarding fibonacci... is that why f(n+1)/f(n) when 'n' is very large tends to the golden ratio? because the lower part of its generating function has same roots that give the golden ratio?

I like that -1/12 reference

I'm still emotional about the -1/12 thing. The whole video should have a trigger warning on it so people like me won't get their feelings hurt.

What is the name of your closing music? It sounds like a C64 SID file. Keep up the great videos, they are a great deal of fun!

5:00 Let's call this number X. Assuming two of the numbers you're raising to the fifth power are just one apart, and we call the smaller one of that pair A, then [the cube root of X] = A^3 + [cube root of [5! * A] ] Not sure where to go from there but having a cube on one side of the equation and a cube root on the other kind of shows how massive the solution has to be.

Proud be an Indian😘😘Ramanujan sir

158_4+59_4=635318657 134_4+133_4=635318657 :D Trial and error ftw!

Since I didn't see it here, from his sub-reddit: For those looking for the book: Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work (by G. H. Hardy) ISBN-10: 0821820230 ISBN-13: 978-0821820230

Realy ramanujan great mathematician....realy genious

How many other Ramanujans have there been throughout history who were never discovered?

1729 views! What a coincidence

Thank you for adding a link to that Mathologer video. Is this the first time it is linked in Numberphile?

11:36 : I cracked up at that moment...you're a genius ! xD

"And everyone gets very emotional.." I died LOL

"you get a negative twelvth out the other side" love that quote XD

I saw you on TV the other day talking about big planes

Found one sum of fourth powers on my own: 133^4 + 134^4 = 635,318,657. Had to look up the other, though, because the first one was based on a lucky hunch and I wasn't willing to check all the possible combinations. (For those wondering, it's 59 and 158.)

Out of copyright - not available online... what is this?! Google. What do we pay you for?! Why is this not in your library?! Just kidding; love you Google... but still, please???

I wish I could imagine what the Generating Function for all prime numbers would look like...

Thanks your explanations and presentations are pretty good .A month ago you posted a problem dealing with A_4 sheet paper problem .Could i know the answer for that puzzle sir

A valid Explorer of the seas, of numbers. For the Amateur Observation, a comparison of Euler's Intuitions and Ramanujan's techniques would be interesting. What exercises do they use to analyse the mathematical problems. It's as if the Measurement Problem, (based on Real Number consistency), is turned insideout and they have "done 10,000hrs" of mental practice in imagining A Quantum Bonding Chemistry of Numeracy.