Ramanujan, 1729 and Fermat's Last Theorem

[@thesavantart8480]

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.

[@openlink9958]

A genius discovering independently anothers genius' work!

[@yanwo2359]

"... 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. :)

[@standupmaths]

I try to aim for two laughs per viewer per video.

[@ΙωαννεςΦαασεν]

206.356*2 so far. I wonder what the ratio of actual laughs to your aim is; I bet you're reaching >1.

[@4124-u7o]

I

[@ReedHarston]

Sadly, I didn't know the back story for -1/12 yet and had to look it up, so I didn't get as big of a laugh at that part of the video on my first time through, but I did laugh at it later. ;) And the end made me just bust out. :D

[@spdcrzy]

Well that explains MUCH more succinctly than Cesaro summations why Ramanujan was able to come up with 1+2+3+4+...= -1/12 lol.

[@bsebire]

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

[@echoromeo384]

Exactly. What a beautiful number.

[@jimf2525]

It’s 16:47 because to get to 17:29 requires life, universe, and everything.

[@איתיסמואלוב]

@@jimf2525 nice!!

[@TheProGam3rHD]

1729 Likes on this comment everyone. Let's go.

[@shawn980]

nice

[@prateekgupta2408]

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

[@evilcam]

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.

[@PauLtus_B]

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

[@leftysheppey]

+PauLtus B Yes. Just yes.

[@PauLtus_B]

I so hope Matt reads this. The Parker Square is something to be proud of.

[@tehjamerz]

😂😂😂 well played

[@themanagement69]

hahahahaha, win.

[@ferretyluv]

We will never let him live that down.

[@DamaKubu]

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

[@jassi9022]

Danielius 😱

[@yashgaikwad7516]

He would probably be a doctor or an engineer

[@gauravsingh3007]

@@yashgaikwad7516 As a fellow Indian, I think I know what you mean. Haha...

[@dekippiesip]

To really comprehend how sick his achievement was, consider this: Andrew Wiles, the already extremely smart guy that proved Fermat's last theorem, spent 7 years trying to prove it using loads of techniques and concepts that had been developed in the 20th century. Ramanujan on the other hand lived in the beginning of the 20th century, so he had independently invented some of those very same concepts himself, and already tried using them in his proof!!! This means he was on the right track, and considering these where scribles he made in a notebook during travels he made, he clearly wasn't investing nearly as much time into proving it as Wiles while at the same time having the disadvantage that he had to come up with stuff that was already well established in Wiles time. It's like someone from ancient Greece who knows absolutely no calculus(cause it wasn't developed) and only has a solid background in classic geometry coming close to proving advanced theorems in calculus by individually developing a large part of the machinery of the subject that took hundreds of years and a lot of people to develop! That's just completely insane and of the charts, and what I would imagine only a highly advanced alien with a much larger brain than a human to be able to do.

[@amberheard2869]

I assume your proper education means he can study mathematics all the day if this was possible he will become insanely smart plus if he had lived long live more and more mathematicians will loosd their probability of doing their own contribution.

[@jamesl8640]

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

[@dixitkhanal8885]

he would be out of content, because he is the content

[@jacobschiller4486]

He would be the longest-lived person ever at 133 years old.

[@pvs_np]

@@jacobschiller4486 Damn, it doesn't even sound thaat old.

[@rishabhlakhara3396]

Lol

[@vanshsinghai8990]

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 .

[@devonmatthews6443]

Oh . really.. Yeah...

[@TrimutiusToo]

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.

[@austinbryan6759]

He was dissing numberphile not Ramanujan.

[@TheEvilVargon]

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

[@BelialsRevenge]

+TheEvilVargon ok at least 1 /1103 people know of this by now. if you add the 2 like to this its already 3/1103. Sorry

[@otakuribo]

Ramanujan gave it a go! 👍

[@_thank_you_]

THIS

[@markoftheland3115]

that comment is quite a Parker Square of what i was trying to find in his channel

[@alexsheehan9290]

I actually loled at this

[@miabussell0229]

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

[@joelhaynie5056]

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,!

[@xylophone897]

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

[@mrZbozon]

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

[@Arkalius80]

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.

[@monicasingh4348]

It's conceptually wrong

[@backslash8874]

Nope, it's not that great.

[@micahyang3395]

"Not by a physicist" Shots fired

[@aasid2446]

Dream intensifies

[@ariel.l.borrero]

So mathematician = sith lord and physicist = jedi?

[@xolanichristopher2002]

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

[@AK47_414]

Study them,they're actually interesting

[@hanniffydinn6019]

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

[@austinbryan6759]

He was smart sure but I think you're misusing the word "easy" here. Even Euler wasn't able to prove many things he tried to do, so why would this guy be able to prove something that took many more forms of mathmatics made and proved by many other people across many years?

[@xqt39a]

Hanniffy Dinn Easily? I don’t think so. He was on the right track and he could have cracked it in maybe 10 years. Recall that Wiles had failed and needed help by another mathematician in the end

[@xqt39a]

Austin Bryan, Right, Although Ramanugen was on the right track with elliptic curves, Wiles took on the project only after elliptic curves were linked to Modular forms which were invented after Ramanugen’s time.

[@youneverknow5555]

kzbin.info/www/bejne/g5_WlqCZYtqDh9U Ramanujan is such a guy who accomplished several new theorems by his own all before 22 years. He was one who freaked his foreign teacher when he straight away gave the final result without the in between steps. Given that he had displayed the ability to give the results in second we are not the ones to do the astrology of "he will accomplish it within 10 yrs ..12 yrs etc..''.

[@backslash8874]

@@youneverknow5555 I agree with you. Ramanujan was a mathematician of the highest caliber. And look at how George E. Andrews states the meaning of Ramanujan in 2014, kzbin.info/www/bejne/r5CTf6iFd7OimM0 . Only mathematicians will know.

[@yashgaikwad7516]

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.

[@Rararawr]

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

[@standupmaths]

That is some great spreadsheet work! I admire your dedication to conditional formatting.

[@Rararawr]

Being able to see what cells are above and below the target really helped to find it. It made a really nice curve

[@barakeel]

I think the point was to solve it by hand using Ramanujuan method.

[@balern4]

​@@barakeellol

[@therealpanse]

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

[@wolken_bruch]

i was looking for this comment! :D

[@therealpanse]

BibiCookiecat you're welcome

[@EGarrett01]

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

[@standupmaths]

+EGarrett01 No problem! If I had a better set-up I'd scan the whole thing.

[@johnchessant3012]

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

[@nrpbrown]

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

[@BeatSyncBytes]

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

[@blueredbrick]

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

[@birdy_coolbeans]

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.

[@tomraj]

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

[@MrCrisC]

65601^3 + 67402^3 = 83802^3 + 1

[@DrZaius3141]

I was glancing at the screen when he mentioned that example. ODD^3 + EVEN^3 = EVEN^3 -> nope.

[@KenBellows]

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

[@AntonioBarba_TheKaneB]

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

[@joemailloux]

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

[@fakjbf3129]

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

[@damienw4958]

I concur

[@Libya4LY]

Please Matt! And go all the way and make it comprehensive, don't make it a Parker Square video...

[@Xclann]

11:37 good one!

[@midwinter78]

"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."

[@ishrirampersad8809]

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

[@rishsharma16]

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.

[@doctordrunkenstein.9448]

Our uncultured bloody film industry care of all the uncultured and spoiled stuffs like Varun Dhawan and Alia Bhatt etc.

[@arnavverma2461]

So so so true 😭

[@rubbersidedown7992]

India didnt even give him a degree

[@ffggddss]

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.

[@pushbutton8548]

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.

[@MyJuicehole]

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.

[@CharlesPanigeo]

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

[@madmax797]

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..

[@luc8043]

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

[@Indianfusion253]

You a my favourite mathematician

[@RQLexi]

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

[@_wetmath_]

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

[@jordibear]

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

[@sitearm]

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

[@bibin3458]

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

[@chanjoshua3539]

Najib in India Asean

[@miladirani4313]

I m too hardy is great man in my country iran most people is gealous to talented guys

[@suresh1957]

Brilliant ! Simply love the way you explain it Sir !

[@BrownHairL]

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

[@humblebeginnings6494]

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

[@Kazutadashi]

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

[@ace.of.space.]

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.

[@standupmaths]

+CraftySunshine Yes, we're almost indistinguishable.

[@MrPhysicist]

@@standupmaths lol..nice joke if youtuber seems to ramanujan..then i am similiar to einstein

[@MrPhysicist]

@@standupmaths r u crazy?? my fellows are doing the same thing and publishing research papers doesn't mean they are Einstein or Newton......they are class apart like Ramanujan..an ordinary mathematician can't be Ramanujan...120 years ago..India was under british rule..no internet..no much education ,no schooling and degrees like u have now

[@MrPhysicist]

@@standupmaths u shouldn't compare urself with nobel man.... u r just a youtuber and a gud mathematician if u r equivalent to him..then call me Einstein

[@backslash8874]

@@MrPhysicist Calm down. He is joking.

[@ByksterK]

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!

[@brohanime]

that's an interesting way of approaching the problem. i wrote a couple lines of code and my computer spit out the answers in milliseconds.

[@vonhart8347]

@@brohanime can you share the code

[@ferretyluv]

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.

[@Sinom.]

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

[@Pom2.0]

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

[@woud3404]

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

[@lweyhacker5557]

You are awesome matt

[@standupmaths]

You're awesome as well!

[@SergeofBIBEK]

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.

[@tggt00]

+SergeofBIBEK Can you explain to me why are you emotional about that thing, I never understood why the fuck people get mad about it, the video just made me love math even more.

[@SergeofBIBEK]

tggt00 haha, it's just a joke. (And I'm pretty sure Mr. Parker is joking too. ) Though I think the general idea is: You add up a bunch of positive whole numbers and end up with a negative fraction. Both the negative and fraction part makes no sense. And people don't like it when everything they've ever known about the world suddenly isn't true anymore.

[@tggt00]

+SergeofBIBEK You're supposed to be amazed by it, and also why the fuck would you not listen to a professor who knows what he saying? It's not like they're lying to you on purpose. I'm mad because a lot of people act like you.

[@SergeofBIBEK]

tggt00 O_O There's a disconnect between what I wrote and what you took away from it.

[@NoriMori1992]

+SergeofBIBEK Hahahaha. X'D I feel the same way when I see it, and likewise when I see the word "Riemann", or that ζ(s) thing. XD

[@99bits46]

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.

[@leadnitrate2194]

No, Ramanujan was an actual mathematician, and a great one at that. Not someone who had just memorised a few algorithms to multiply numbers, which I'm sure is hard, but not quite on the level of Ramanujan. The fact that he could visualise numbers was just an added bonus.

[@LazsalzariRomnzevroskki]

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.

[@TheGremlin012]

According to me mathematics is the language of the Universe, and as with all languages all you need to do is persist. Keep going. :)

[@zozzy4630]

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

[@willyengland]

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

[@manaoharsam4211]

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.

[@smoorecrux]

I spent the whole video swearing I could smell that very old book smell.

[@standupmaths]

So could I!

[@namantenguriya]

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

[@TheOneTrueIgnus]

If Matt had made the video just a little bit longer, it could have been 17:29 long :)

[@MaterClaritas]

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

[@ChrisChoi123]

+TertiusIII Great idea. It would be funny to see Jeremy argue with Matt about some of the sins like he did with neil degrasse Tyson.

[@NoriMori1992]

+TertiusIII That's a brilliant idea!

[@CraftQueenJr]

TertiusIII and with matpat.

[@Czeckie]

cinemasins is garbage

[@Memington]

-1/12 is my spirit animal! How dare you insult it?!?!?!

[@goodboi01]

lol its mine too

[@standupmaths]

I'm, ah, insulting the people who insult -1/12?

[@Memington]

ooooh, I thought you were dissing -1/12. I get it now. Good to know my spirit animal remains unmocked.

[@goodboi01]

+standupmaths those people should go to the roots of thier negative minds....oh wait...

[@binodchoudhary8877]

Realy ramanujan great mathematician....realy genious

[@Dr.Pepper001]

If Ramanujan could have lived to be 90, imagine what he would have left us.

[@KoenZyxYssel]

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.

[@dosluke]

will you do a separate video on generating functions please?

[@dosluke]

Ryan O'Farrell this isnt a formal paper, and thus 100% perfect syntax (not gammar) doesnt need to apply. Yeah thats a syntax error not a grammatical error..

[@panda4247]

+Ryan O'Farrell: The Internet is a proper noun (it's a name of the network), and as such should be written with capital "I". If you are a grammar nazi (especially when it's about such a minor issue, that is not changing the sentence's meaning - correcting people in "your vs. you're" situations is somewhat different), at least be proper about your own writing.

[@Smitology]

@@ryanofarrell186 Has that error changed your comprehension of the sentence?

[@bartdavies5038]

he died from malnutrition and was disrespected. Good movie.

[@DanTurismo1]

0:45 ...c'mon Matt. We're all maths people here

[@sabyasachisanyal4865]

I am non maths person and I liked it

[@Dusan994Q]

I like that -1/12 reference

[@lafondawilliams]

I loved the film

[@ledermueller]

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!

[@Jackthefounder]

the movie was great

[@RobertWF42]

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

"Interpreting infintie as infinite" Best bit of the vid ;-) ;-)

[@കാർത്തിക്വിജയൻ]

Proud be an Indian😘😘Ramanujan sir

[@joea8426]

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

[@U014B]

If x^2 is "x squared" and x^3 is "x cubed", wouldn't that naturally make x^4 "x tesseracted"?

[@whitherwhence]

"x minuted"

[@Ucedo95]

I once heard students say "x hypercubed" at a national math competition.

[@NoriMori1992]

+Noel Goetowski I literally thought the same thing to myself!

[@suspendedsuplexchannel1000]

Noel Goetowski good job you are a mathematician

[@sekarperumal6287]

Ramanujan India's Pride....

[@Will140f]

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

[@standupmaths]

It was a great writer, if a little grumpy at times. I recommend his book A Mathematicians Apology as an interesting read.

[@Will140f]

standupmaths thanks Matt! Always looking for recommendations.

[@minerscale]

This is what I love about maths. If I set up a computer and looked, I could be the first person to find the lowest 5th power taxi cab number.

[@jeffreybernath6627]

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

[@standupmaths]

That's the spirit!

[@lawrencecalablaster568]

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

[@GuerrillaSauce]

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???

[@deproissant]

Is that a shot towards Henry of minutephysics at the end of the video?

[@KennedyDaniels]

1729 views! What a coincidence

[@3Designlab]

Kennedy Daniels 👍

[@Mortiis558]

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?

[@tejasdeepsingh456]

It's actually a great film

[@joemcclinton9001]

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

[@enolastraight577]

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

[@paologalliani4172]

Ramanujan said that a goddess gave him the solutions...but I don't understand why the Goddess did not give him for example the solution for the fourth powers...so I am bit skeptical that he was actually inspired by a Goddess