How they found the World's Biggest Prime Number - Numberphile 

+mrkarlhey not really unless you're really into that stuff and can compare it to other values for random activities

+Wabadabadoe It's only a ream and a half per copy

+mrkarlhey It wasn't found with extremely advanced computers rather 800ish very average computers months.

+mrkarlhey It wasn't found with extremely advanced computers rather 800ish very average computers months.

+mrkarlhey It wasn't found with extremely advanced computers rather 800ish very average computers months.

1:59

Stop making Parker squares a thing! -Matt Parker

@computer hi matt

What's a Parker square?

@@createyourownfuture5410 "parker square" refers to an infamous mistake by mathematician Matt Parker

eZ

+donbasuradenuevo Use binomial expansion.

3,848,889,888 3,848,889,890 3,848,888,897 whoops gotta start over

I predict you just tested 2^7-1 already.. Didn't u??

@@monicarosas264 The mathematician William Shanks took 15 years to calculate 707 decimal digits of pi, but he made a mistake at the 528th position.

try hunting for more even primes

Isn’t one also a prime or is it not considered a prime because it’s also a square

@@gatlinggun511 1 is not a prime due to the fact that it only has 1 factor. a prime number has 2 factors, 1 and itself

@@gatlinggun511 1 isn't considered prime because it would break the fundamental theorem of arithmetic, as they would then have infinite prime factorizations due to being able to have an arbitrary number of ones in the factorization, so we decided to just not include it as prime (among other reasons, but this is a big reason why)

It’s infinity right?

+Thingsandstuff Now it's 4, 5, 6, 1, 2, 3, 7 xD

+Thingsandstuff thanks

+Thingsandstuff No they go: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282

Was there a joke here I missed?

+E.B. White George Lucas

+Harry Tsang Like all Mersenne numbers

+Harry Tsang Yeah, I am so smart that I learned it by heart!

+TheJman0205 00110000 00110001 00110001 00110000 00110001 00110001 00110001 00110001 00100000 00110000 00110001 00110001 00110000 00110001 00110001 00110001 00110000 00100000 00110000 00110001 00110001 00110000 00110000 00110001 00110000 00110001 00100000 00110000 00110000 00110001 00110000 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110001 00110000 00110001 00110000 00110000 00100000 00110000 00110001 00110001 00110000 00110001 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110000 00110000 00110000 00110000 00110001 00100000 00110000 00110001 00110001 00110001 00110000 00110001 00110000 00110000 00100000 00110000 00110000 00110001 00110000 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110001 00110000 00110001 00110001 00110001 00100000 00110000 00110001 00110001 00110000 00110001 00110001 00110001 00110001 00100000 00110000 00110001 00110001 00110001 00110000 00110000 00110001 00110000 00100000 00110000 00110001 00110001 00110000 00110001 00110000 00110001 00110001 00100000 00110000 00110001 00110001 00110001 00110000 00110000 00110001 00110001 00100000 00110000 00110000 00110001 00110000 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110000 00110000 00110000 00110001 00110000 00100000 00110000 00110001 00110001 00110000 00110001 00110001 00110001 00110001 00100000 00110000 00110001 00110001 00110001 00110000 00110001 00110000 00110000 00100000 00110000 00110001 00110001 00110000 00110001 00110000 00110000 00110000 00100000 00110000 00110000 00110001 00110000 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110001 00110000 00110001 00110001 00110001 00100000 00110000 00110001 00110001 00110000 00110000 00110000 00110000 00110001 00100000 00110000 00110001 00110001 00110001 00110001 00110000 00110000 00110001 00100000 00110000 00110001 00110001 00110001 00110000 00110000 00110001 00110001 00100000 00110000 00110000 00110001 00110000 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110001 00110000 00110001 00110001 00110001 00100000 00110000 00110001 00110001 00110000 00110001 00110000 00110000 00110001 00100000 00110000 00110001 00110001 00110001 00110000 00110001 00110000 00110000 00100000 00110000 00110001 00110001 00110000 00110001 00110000 00110000 00110000 00100000 00110000 00110000 00110001 00110000 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110000 00110000 00110000 00110001 00110001 00100000 00110000 00110001 00110001 00110000 00110001 00110001 00110001 00110001 00100000 00110000 00110001 00110001 00110001 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110001 00110001 00110000 00110000 00110001 00100000 00110000 00110000 00110001 00110000 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110000 00110000 00110000 00110000 00110001 00100000 00110000 00110001 00110001 00110000 00110001 00110001 00110001 00110000 00100000 00110000 00110001 00110001 00110000 00110000 00110001 00110000 00110000 00100000 00110000 00110000 00110001 00110000 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110001 00110000 00110000 00110000 00110000 00100000 00110000 00110001 00110001 00110000 00110000 00110000 00110000 00110001 00100000 00110000 00110001 00110001 00110001 00110000 00110000 00110001 00110001 00100000 00110000 00110001 00110001 00110001 00110000 00110001 00110000 00110000 00100000 00110000 00110001 00110001 00110000 00110000 00110001 00110000 00110001

01001101 01000100 01000101 01110111 01001101 01010100 01000001 01111000 01001101 01010100 01000101 01100111 01001101 01000100 01000101 01110111 01001101 01000100 01000101 01111000 01001101 01010100 01000101 01100111 01001101 01000100 01000101 01110111 01001101 01000100 01000101 01111000 01001101 01010100 01000101 01100111 01001101 01000100 01000001 01111000 01001101 01000100 01000001 01110111 01001101 01000100 01000101 01100111 01001101 01000100 01000001 01111000 01001101 01000100 01000001 01110111 01001101 01000100 01000001 01100111 01001101 01000100 01000101 01110111 01001101 01000100 01000001 01110111 01001101 01010100 01000001 01100111 01001101 01000100 01000101 01111000 01001101 01000100 01000101 01110111 01001101 01000100 01000101 01100111 01001101 01000100 01000101 01111000 01001101 01000100 01000101 01111000 01001101 01010100 01000001 01100111 01001101 01000100 01000101 01111000 01001101 01000100 01000001 01110111 01001101 01000100 01000101 01100111 01001101 01000100 01000101 01111000 01001101 01010100 01000001 01110111 01001101 01010100 01000001 01100111 01001101 01000100 01000101 01111000 01001101 01010100 01000101 01110111 01001101 01000100 01000101 01100111 01001101 01000100 01000001 01111000 01001101 01000100 01000001 01110111 01001101 01000100 01000001 01100111 01001101 01000100 01000101 01111000 01001101 01010100 01000001 01111000 01001101 01000100 01000001 01100111 01001101 01000100 01000101 01111000 01001101 01000100 01000101 01110111 01001101 01000100 01000001 01100111 01001101 01000100 01000101 01111000 01001101 01010100 01000001 01110111 01001101 01010100 01000001 01100111 01001101 01000100 01000101 01111000 01001101 01000100 01000001 01111000 01001101 01000100 01000101 01100111 01001101 01000100 01000101 01111000 01001101 01000100 01000001 01110111 01001101 01000100 01000101 01100111 01001101 01000100 01000101 01111000 01001101 01000100 01000001 01111000 01001101 01000100 01000001 01100111 01001101 01000100 01000001 01111000 01001101 01000100 01000001 01110111 01001101 01000100 01000101 01100111 01001101 01000100 01000001 01111000 01001101 01000100 01000001 01110111 01001101 01000100 01000001 01100111 01001101 01000100 01000001 01111000 01001101 01010100 01000101 01110111 01001101 01010100 01000001 01100111 01001101 01000100 01000101 01110111 01001101 01000100 01000001 01111000 01001101 01000100 01000001 01100111

+Abandoned Void 01000010 01101001 01101110 01100001 01110010 01111001 00101100 00100000 01110100 01101000 01100101 01101110 00100000 01000010 01100001 01110011 01100101 00110110 00110100 00101100 00100000 01110100 01101000 01100101 01101110 00100000 01100010 01101001 01101110 01100001 01110010 01111001 00100000 01100001 01100111 01100001 01101001 01101110 00101100 00100000 01100101 01101000 00111111 00001101 00001010 00001101 00001010 01010000 01101111 01101001 01101110 01110100 01101100 01100101 01110011 01110011 00101110 00101110 00101110 00100000 01000010 01110101 01110100 00100000 01101001 01110100 00100111 01110011 00100000 01100110 01110101 01101110 00101100 00100000 01001001 00100000 01100111 01110101 01100101 01110011 01110011 00101110

E

They do.

He'll grow.

Can confirm, I saw a technician test my PC with it.

I hope you caught it

You have an exceptional mind.

Don't see how it could've thrown an exception, integer overflow maybe

Oh is that what that sound was?

All even numbers can't be primes expect 2

+t0f0b0 and in binary too^^ ...and quaternary ...and senary Cool, works in 4 distinct bases :D ...But it's not a Mersenne prime >:(

+Benjamin Philipp Wouldn't it be prime in an infinite number of distinct bases as 11 would simply whatever base you choose plus 1?

Jordan Lynch Of course... :D Somehow I was focused on decimal so much that I thought "well, the 'value' of 11 won't change in higher bases" as if it was a distinct digit, making higher bases irrelevant. That's what I meant by "distinct bases": "Eleven is still going to be eleven in bases higher than 10 and will always be a prime" - neglecting the fact that eleven would look like "Ɛ" for example.

I can count to potato!

11 has four factors: 1, 2, 5.5, 11 Trololoollooll jk jk jk jk J U S T K I D D I N G J US T A B R A N K. P R O

Like pharmaceutical companies. They make less from curing disease than by treating it.

Are you the guy who believes the thing that they're withholding cancer cures too?

Lucas Ng yes and no. "Yes" is Companies wont produce any product they think unprofitable. if they could, the process would be secret and the ingredients be partly secret too, so we dont know if there was any better way possible. "No" is hardly found a company invest in such researches, investing in cancer research is super risky, so public laboratories run by the goverment will do this job, and yes, they will publish everything.

publish a prime every month. there will be many more pages each time so it will cost more and more. do not publish the pattern if you're able to find a prime each month or anyone will do it

A company that had the cure for cancer would make an untold fortune from it and would never withhold it. It wouldn't be relevant to them that other companies would no longer profit from treating the disease. Government agencies are run by people who have a vested interest in keeping their funding going on and on forever. There is much mere accountability in private concerns because you only get paid for what you actually accomplish an what you can market. Public (government) concerns exist to milk more and more research money, which would end as soon as they were to find a cure.

It’s in the video

Interesting seeing you here 4 years later

@@eboone why is this video suddenly recommended

@@_wetmath_ might be my fault... I was binge watching numberphile a couple weeks ago, found this one and made a few comments on it. From what I could tell, the last reply was a year before my comments. Just a guess though.

Wait *what?!* What's my favorite music theorist doing here?! Ps. I'm on the spectrum too

Fish Nugget it’s ~ 2 billion billions. Maybe thats why

David - Yea prop lol. But saying a quintillion is better

@@blue9139 lol

But he was back at it again when he turned 7.

Zed dash. silly joke when he. was. 6 he was perfect.

Jayden Tan It wasn’t a 5-year old 5

Zed dash. This entire thread delivers.

Why is 6 afraid of 7? Because 7 8 9.

+TehDragonGuy Why?

+David Greydanus because

+TehDragonGuy What are you going to do when the next largest prime is discovered?

+allyourcode Chop down even more trees.

I want to know how many tons of coal need to be burned for GIMPS to find the next largest prime.

Why?

+Alan Douglas do you have an idea of how big that number is? It's like trillions of digits long (maybe even bigger).

+Alan Douglas do you have an idea of how big that number is? It's like trillions of digits long (maybe even bigger).

+DavidRussell323 was just thinking that, I wonder if the chances of a 2^Mersenne Prime - 1 are more likely the answer is a prime.

+VivaFeverFifa 2^74207281-1 contains 22,338,618 digits. 2^10^22,338,618 would be something like 10^22,338,617 digits long.

Then it's wrong

+Clowen00 TIL Infinity fits in three volumes.

+Belleren savage

That's only because the number is too big for the calculator to calculate

Epic Wolf Oh really? That sounds actually plausible, didn't thought about that! ;-)

He's so brilliant 18 + 29 is fiftyyyyishhh.... ahh 47 !😂😂

Gopher it.

Or at the bank, to ascertain the ruling prime rate?

If they don't give you a prime number of ribs, send it back

@@bananya6020 lol

Side splitting humor 😐

+scozio Gotta make it base 36. Thou has forgotten the numbers.

+Keroia Just encoding

+scozio it says 'illuminati'

scozio it would say "youhaveallbeenfooledforsolongtherealenemyisthemartians7902g4h8kkkkkk1" somewhere in there

scozio base 26?!?!?

+Sam Harper ?

+Albert Chan You are so clueless about how jokes work

I dont unsfdder4y3wfhwhy5

XD

Maybe "the world's" because an alien species couldve found a bigger one already.

+Oddtwang of Dork Testing the primality of smaller numbers will take a few seconds at most (probably not even a tenth of a second for a number of the order of a few billion.)

The real question remains unanswered

The proof is beyond the scope of this video

Just math. You could ask that question about the simplest algebra and end up in a massive loophole of confusing proofs

The one that can answer that is 3blue1brown..

@@050138 The proof is left as an exercise to the viewer.

10:46

this is hilarious, he sounds absolutely drunk XDDD

“for a dAay wE g- gAt sAmthing tOo dOo with oUr côMpüÜtèrs”

that's a lot of numbers!

The biggest number SO FAR that anyone's bothered proving is prime. As shown by the fact that this number is nowadays nowhere near the biggest any more

-1 (_ _)

I mean that's a really low bar to set though.

Neel Modi I know for a fact 2^74207281+1 is not prime. It is divisible by 3

If 2^74207281-1 is prime, and 2^74207281 is an even number with no factor of 2 - then 2^74207281 must divide by 3.

@@theleftuprightatsoldierfield Ohh man u spoiled his attempt for a joke😂😂😂🤣🤣🤣 Anyway that was perfect ....

Jajja, two year ago discovered other Mersenne prime with more than 24 millons of digit. The gigant Merssenes primes numbers discovered in the last 20 years was a distributed work of millons of personal and server comouter around the world using the Primes95 application conected in the GIMP project.

+ytYAEeLxmEYb As compared to infinite...

+Red Sniper Well, infinitesimals are actual things, used in calculus, and that is definitely not one, but yes compared to infinity it is a small number.

Draevon May Well thanks for info but i already know that... :P

+Red Sniper Assumed you did. I'm used dealing with highschoolers. Not the brightest bunch of people 😆

Draevon May XD

+A V Sandi Nack $2^74,207,281 -1 USD

+josiah O'Neill smart aleck :)

+YipYapYoup I thought your dollars were bimetal. :)

+A V Sandi Nack commenting for captain

+A V Sandi Nack mathsgear, maybe

they've checked up to around 10^18. I think they've gone a bit further but haven't gotten to 10^19 yet.

+Cruzer Since doing so is easiest based on a sieve, once you know the list up to that point, the next is easy to find. Too hard to define a specific point for that reason, and thus little reason to even search.

+xunile1 But remember they (Gimps) are only checking one type of prime - there may be other types which haven't been checked up to that range yet.

+Gordon Taylor the numbers that GIMPS is searching for are way larger than 10^18, all primes up to 10^18 have been found.

Cruzer i can check the largest ones conceivable factors 2=no 3=no 6972=no We know it is a prime, so we know all of its potential factors aren't it's factors

Always nice to meet a fellow traveler, chasing fireflies through the fields of math!

Rik Schaaf 356787 42157899865323466755443278887765556789000009887665433457788839387474738289254333564215789986532346675544323585858696969699988665323567898531245685652413131453645

@@janeemmanuel8885 I mean, if you are gona smack your head on the keyboard in order to guess an answer to his question, at least make sure it does not end with digit 5 lol

Sergej - That's hilarious! I love it.

@@sergejkeser7270 😂😂

@@sergejkeser7270 dont forget about the random space 7 digits through

yes

+Sean M would be fun just to hear them struggle with pronunciation

The interesting thing about 239 is that it's the only prime number divisible by 239. So what, you say? Well then, why does everyone keep taking me that 2 is the only prime number: all they mean is that it's the only prime number divisions by 2. But you can say an exactly similar thing about EVERY prime. So what?

@@trueriver1950 There are some other primes of the form x^y-z

The computing power needed to check that would be astronomical.

+SPENCER LEMENS 2^(2^74207281 -1) -1 ftw

+SPENCER LEMENS Sadly, the exponent being prime is only a necessary but not sufficient. Therefor the converse is not true. Example: M11. It's a Mersenne number with a prime exponent: 2^11 - 1 = 2047 but the result is not prime since 2047 = 23 * 89. It's still a good idea to use this prime as exponent and eventually it will be checked at some point ... it may even be prime, who knows -- but it's not guaranteed to be a prime.

+aednil Considering banks use 2048 bit numbers now and modern computing would take thousands of years to break the discrete logarithm problems, it could be a while!

+James Purcell I'm not 100% certain of how encryption works, but iirc, this number is even useless for encryption, because it is so well known. It doesn't sound completely right, but I heard it from various sources.

levolta The most basic variation is RSA, so in RSA I'd tell you (n,e) = (65,7), e is your encryption component, and n is the product of 2 primes p and q. Obvious with a number this small you can see, p = 13 q = 5. Now if you want to find the decryption component, you do e * d = 1 mod (p-1)(q-1) , so 7 * d = 1mod 48. Again as it's a simple example you can spot if d = 7, then 7*7 = 49mod 48 = 1mod 48. So if I give you a message say M = 3. 3^e mod n = C your ciphertext. So 3^7 = 2187 = (33*65) + 42 mod 65 = 42mod 65. So after encryption of a message M = 3, we get a ciphertext C = 42. If you want to decrypt this, you can do C^d mod n, so 42^7 mod 65 = 3mod 65. So a message raised to the power of the encryption component becomes ciphertext. a ciphertext raised to the power of the decryption component is your original message. If your primes become huge (Banks use 1024 bit primes), these numbers are crazy big, and it is computationally infeasible to find the primes P and Q if given the product N. If they can't find P and Q they can't find the decryption component and you're messages and bank details ect are safe. Because this prime is so large it isn't really an issue, and won't be for a large amount of time. Banks at the moment use two 1024 bit prime numbers, to make a 2048 bit product N, this takes current generation computers millions of years of constant computation to brute force. I doubt banks will ever use a prime this large in security based systems such as RSA.

+James Purcell what levolta meant is that if you did decide to use this specific prime, people would already know one of the components, so you already have the job done for you, you successfully decrypted an RSA key the strength of RSA comes from not being able to find the components in useful time, but if you know one of them, the job is already one, it's a linear operation to find out the other one and you can guess the private key.

Pedro Gusmão While that is true, banks don't tell you what either of the primes are, if they started using a 44million bit prime people would get suspicious sure, but as more primes are discovered and none mersenne primes are discovered through things like prime grid theoretically a well known prime could be used, but it would be dangerous as you've said

Correct: We know all the previous primes of various types. For example, whatever the current biggest Mersenne prime is at any time, we likely know all the smaller ones in that series of primes. Ditto for any other family of primes (Sophie Germain, etc...) What we don't necessarily know is all the primes in between those in the various series of easy-ish primes. We know approximately how many there are in any gap, but without testing exhaustively we never can be sure if we have them all. So your question amounts to asking how far has anyone got doing an exhaustive search for primes starting from 2. Apols if you already figured that out

On quantum, probably not. But for classical, yes, because RSA