Hey, that makes sense to me. "Recreational" there means HE finds it fun. I'm a linguist, and I have fun with languages in ways that probably make no sense to most people.
What makes something a recreational problem? Pretty much all of research level maths is recreational in that it has no obvious uses. It's a pretty natural question to ask. Also, proving anything original about the primes is a huge achievement
What's striking about the prime that Matt printed out is how uniform the frequency distribution of the digits is. It's perhaps not all that unexpected, but it's interesting to see it explicitly displayed.
If you think about it, digits of a number are remainders after division by powers of the base. If the number is susficiently large, the frequency of any given digit approaches 1/base.
@@palmomki But they are remainders after dividing by base to some power. 1234 mod 10 is 4, you have your last digit. You divide by ten ignoring the remainder, so now you have 123. 123 mod 10 is 3. Your second to last digit and so on. This is how you can convert numbers to other bases. Pretty simple honestly. As for the second statement. For a random number that statement is true. You have a 1/base chance for each digit to be put in the number. Sure if you pick 111111 it doesnt apply but for big enough random numbers I"m pretty sure it does.
@@palmomki ok, i get what you are saying. But, you get the intuition from what he was saying, right ? I understand it that way : Let k be a natural number, X a random number : X ~ Unif{1,..,10^k-1} We define Xi such as : X = X0 + 10*X1 +... + 10^(k)*Xk Xi ~Unif{0,...9} let a be a number in {0,..,9}: Frequency of a in X in mean 1/k*E[sum(1[Xi=a])] = 1/k *k*1/10=1/10 And this holds for any k, so The frequency of a digit appearing in a whole random number (defined above) is 1/10 (Generalization give 1/b)
They were wrong. You can't have a prime number only made of the digits 7 (exept for the prime 7). Because: 77 divides into 11 777 divises into 111 7777 divides into 1111 And so on. A number made of only the digit 7 would be able to be divided into 7*111111111... So you cant have a prime number only made of 7's (except for the prime 7 of course). This means you: - can't have a prime only made of 0's. - could have a prime only made of 1's (as far as I am concerned) - can't have a prime only made of 2's and/or 4's and/or 6's and/or 8's because they would divide into 2. (except for the prime number 2) - can't have a prime only made of 3's and/or 9's beacuse it would divide into 3 (except for the prime number 3) - can't have a prime only made of 7's. Because of the proof over. Conclusion: If you want a prime number only made of 1 type of digit, the digit must be 1. (Excluding the primes 2, 3, 5 and 7).
I hear 'all 7s' and immediately go 7 * 111111...111 etc. *Edit for a few people:* I'm not saying anything negative about anyone in this video, just bringing to light an error that was made for viewers. I understand completely that it was an on-the-spot discussion and that errors can and will be made, and in no way was I trying to be disparaging. I didn't in any way expect this to get as many likes as it did, so thanks, I guess?
For a guy who won a Fields medal he seems remarkably relatable and down to earth. That combination of intelligence and ability to communicate his work is admirable
@@hewhomustnotbenamed5912 His comment actually has 7 likes now. But I can't possibly deduce if that happened before your original comment exceeded 77 likes or not.
@@lonestarr1490 I can but don't want to. The wayback machine is an online archive of millions of internet pages at different times, but I'm too lazy to check this TH-cam video at different times. You could try it if you want.
Seeing a number that large printed out like that gave me goosebumps. It's obvious but at the same time absolutely mind blowing to see it like that. It's like staring into the abyss.
@@maximilianlorosch936 Assuming 111 had no other factors, it doesn't follow the same pattern as 7 because 1 isn't a prime factor. And if you take 11, then it still doesn't work.
7:50 A number consisting of all 1's is a "repunit" (1 == unit), but with all 7's would be a slightly more generalized "repdigit". BTW, repunits or repdigits can be specified for any base. For example Mersenne numbers are base-2 repunits.
Ok, but here's an idea: if we can similarly prove that there are infinitely many primes whose binary expansion has no zero, that would mean there are infinite primes as strings of ones in binary, which are always 1 less than a power of two, which are Mersenne numbers, which are linked to the perfect numbers... So it would function as a proof of infinite perfect numbers!
guys plsss stop pointing out the same mistake of the 77....77 being divisible by 7 and 11....11. It's already pointed out so many times that I cannot enjoy reading the comment section
7 is the first weird prime. Composite numbers don’t feel random, 2 and 3 are subitizable, and we have 5 digits on each hand and foot. It’s also the first harmonic ratio factor that starts to sound a bit alien. It’s also kinda near to 10/(the golden ratio), while still sharing no factors with 10-intuitively, the significance of this maps to the “random walk” feel of the units digit of every multiple of 7 (in order).
If you take the latest current known prime, and convert it to binary, you end up with a number with only one and zeroes, that will be at least three times as many digits. Another way to do it, would be to convert the prime into the base of itself. That would be 10, which also does not contain 7. At least bases are mentioned.
@@garyz2674 I feel like you don't quite get how bases work. 7 in base 7 is represented as the digits 1 and 0. That doesn't mean it's now the number three higher than 7 and has new factors.
@@OMGclueless In base 7 there sure is a 10, but the symbol 7 is absent, in all otherworldliness. To rephrase my point: would you count something that, by definition, you say does not exist in the context you've postulated?
@ The number seven exists in base seven and it's called "seven", just like the number ten exists in base ten and it's called "ten". What doesn't exist is a digit "7" to denote seven, the number exists just fine.
7 is chosen most often because people see it is the least ordinary number. Remove the evens -> 1,3,5,7,9 Remove the first and last -> 3,5,7 Remove the middle -> 3,7 Pick the second one remaining -> 7. Oddly enough - the 7 of diamonds is a common card people choose when thinking of a card.
3:20 Just curious: does proving that the sum of 1/n for every positive integer n with no 7 in its decimal expansion involve a lot of the Inclusion-Exclusion Principle?
I an fascinated by primes and all the videos from Numberphile that feature primes and James Maynard. I'm also fascinated by interesting facts about primes and the Twin Prime Conjecture in particular. For example you know the Pythagorean Triplets like 3, 4, 5 where there are right triangles with all integer length sides, right? Well, every pair of twin primes is part of an integer solution to a^2 + b^2 = c^2 due to the fact that (n + 1)*(n - 1) always equals a perfect square, so multiplying any Twin Prime pair produces a perfect square and you have a Pythagorean triple on your hands. I think a geometric type proof combined with using imaginary numbers could potentially prove the Twin Prime conjecture. After all, if we ran out of twin primes we'd run out of this class of integer solution to the Pythagorean theorem and we'd run of of rational fractions to fill in the spaces between all those irrational numbers on the number line to express completely unique slopes as angles get smaller if it wasn't true. There can always be a prime just below the square root of a perfect square and there can always be one just above the square root of perfect square. We just need to prove there are infinitely many perfect squares where primes exist in both places. Why not? What would stop them? So this seems "doable" to me. (I wonder if James is reading this...) I also have an idea that primes come in families that form a series where if you only take the primes in a given family, they plot nicely in a graph and the higher up the number line you go, the first element of new families keep appearing. (KInd of a recursive, fractal-like way of generating more primes to fill gaps as needed). That's just a wild hunch and could be baseless, but facts like if a member of the tribonacci series is a prime, the index of that member is always prime lead me to such conjectures. For example, the primes that appear in this series form a sort of family and the indexes that match up to the primes also form a sort of family. Both are subsets of all primes, but what's the underlying connection? If the index is prime, there's no guarantee the member is, but if the member is, the index must be. It's so interesting to me. Why should such a fact even be true? I'd love to talk to James if he ever had some free time to do it..I'm a total prime number nerd and love them.
Wonder what's on the brown paper that is framed on the wall behind him? I was expecting "Dr. James Maynard, maths professor at Oxford" to be an old, distinguished, bespectacled, balding, grey-bearded cat. Turns out he's a KID!
I assume once you get to sufficiently large #'s of digits, each digit has a roughly equal distribution of being any number overall (when averaged out over the whole length of the number) Your example of the 3 volume number follows this roughly even. Thus, the likelyhood of the number having no 7's is basically just 1/10 per digit, and then multiplied to the exponent of however many digits there is in length, and this gives the rough distribution of primes at that digit length that we would expect to contain no 7's, (or any other chosen number
It’s crazy to think that this number, being over 10 million digits long, is merely somewhere between G1 and G2 of Grahams Number (G64)… G1 being 3↑↑↑↑3 (which is equal to about 10^107,000,000,000,000) and G2 being 3↑↑↑↑3 to the power of 10^107,000,000,000,000). Now repeat this hyper-exponentially only 62 more times and finally you’re at Grahams number!!!!
Wrong, G1 is 3^^^^3. Just 3^^^3 is already far too big to fit in the universe, as 3^^^3 evaluates to a power tower of 3s that's 3^27 high. You'd have to write the power tower of 3s from here to the sun, and then evaluate it from top to bottom. G1, or 3^^^^3 is 3^^^3^^^3, so you then have to evaluate 3^^^3 layers of power towers which are themselves power towers. G1 is vastly bigger than you think.
Wait literally the *only* thing I care about is Numberphile videos with James Maynard. What the hell, TH-cam. Were you ever going to tell me about this?
I love watching videos of Bouncy Dude. You can tell the wheels are spinning. Or genius pistons maybe...I don't know. Whatever..... when you see it, things are fix'n to get solved!
All sevens: it is divisible by 7 and the number made all of 1s with the same number of digits, so definitely not prime! Only works with only 1s to get (possibly) a prime.
@@smrusselkabirroomey7396 It is easy to miss that when you get asked it on the spot. I know I have done it many times, thinking about something later and realizing I missed the obvious! Prof. Maynard had a lot to think about, making sure he got all the bits he wanted to talk about, in a decent order, clear and concise, etc.
I wonder if there's an infinite number of primes made up of repeating 1s. (Or, to put it more rigorously, are there an infinite number of primes that can be defined as the sum from 0 to n of 10^n?) edit: im dum dum who didnt watch the video through
I'm really glad you showed the clip at the start of the largest known prime and how evenly the digits are distrubuted within it. Really puts into perspective how uncommon a prime with absolutley zero 7s in it would be, and yet there are still infinitley many of them.
The somewhat hard proof might mean that its not so obvious this should be the case. If the probability goes to down to infinity to get a prime number with some property, but you have infinite "random" numbers still to go, is it guaranteed you will always have infinite numbers such as these? Maybe Trump knows
@@codycast Not the single primes themselves, but the complete set. It is completely deterministic but has MOST of the characteristics of a random distribution. Most of the he few (non-trivial) patterns that we now of, are still a mystery to us. In the vid, it was shown that the sum of the inverse prime numbers diverges, but JUST barely. Prime factorization is the base for contemporary cryptography, the Zeta function, which is basically prime factorization in the complex plane, contains one of the biggest unsolved problems in current math. Primes pop up in every area of math and are so fundamental that even natural evolution has stumbled upon them several times as a solutions to different problems.
Xario Withoutalastname fair enough. I guess I just don’t know enough to have a proper appreciation. I wonder why the video didn’t show the largest prime number known without a 7
I think there was a mistake here. The only repdigit prime possible is all 1s, repunit numbers. A repdigit number with all 7s is always divisible by 7. Still, proving there is infinitely many repunit primes would be super cool.
Hey James, congratulations for winning the Fields Medal for 2022 for your contributions to Number Theory! I was wondering why old Numberphile videos that I've already watched are showing up in my TH-cam feed all over again ... and now I know! Nevertheless, I'm gonna re-watch them all over again.
James is exploring what he loves, on the frontier of human knowledge, with such humour and enthusiasm, and who knows where this research or the techniques being developed could lead. Great video, thanks !!
7777.... will always be divisible by 7 You'd think that would already be considered, and dismissed when you're already having the 1111.... issue that you've considered.
You can't have all 7s because it's divisible by 7; specifically a number that's 7 repeated n times is equal to 7 * 1 repeated n times (responding to the discussion @ 7:30 )
rahowhero X yeh I could do, there’s nothing stopping me. You can just walk into the maths institute and into a lecture, don’t have a register and you don’t need to scan your card at the door or anything.
All single digit numbers can be divided by that digit. One way to see this is consider they all can be multiplied by 10, or 100, or 1,000, etc. Numbers composed of only the digit 1 can potentially be prime because dividing by the digit in question (1) doesn’t count.
7:00 As for the repunit 1, it is not only that multiples of 3 don't go. It can only be a prime if its number of digits is a prime. For example if n=(10^14-1)/9=11111111111111 consisting of 14 digits, you can write n=11×1010101010101=1111111×100000001=11111111111111, because 14=2×7.
idk if i misunderstood this, but he also said that only 1 and 7 are possible to use for repunit primes, but won't any number with all 7s be divisible by 7? or do you exclude the number itself when considering whether its a repunit prime
![[TH] FS vs BME - VCT Ascension Pacific - Lower Bracket Final](http://i.ytimg.com/vi/wnygqsR2Fig/mqdefault.jpg)
The guy just got the Field’s Medal! Congratulations sir 👏🏻
Exactly. He's so much smarter than he seems, since he's trying to explain math in a way us mere mortals can understand.
Feels like a recreational problem...
Writes out a proof spanning 70 pages. Absolute mad lad
Well it helps optimazing the search for primes ... At least it Shows a way that is Not usefull for optimizing the search
Hey, that makes sense to me. "Recreational" there means HE finds it fun. I'm a linguist, and I have fun with languages in ways that probably make no sense to most people.
So many number theory proofs turn out to be really important. Large prime numbers are super important for cryptography.
@@ESL1984 The monster?
What makes something a recreational problem? Pretty much all of research level maths is recreational in that it has no obvious uses. It's a pretty natural question to ask. Also, proving anything original about the primes is a huge achievement
What's striking about the prime that Matt printed out is how uniform the frequency distribution of the digits is. It's perhaps not all that unexpected, but it's interesting to see it explicitly displayed.
If you think about it, digits of a number are remainders after division by powers of the base. If the number is susficiently large, the frequency of any given digit approaches 1/base.
@@palmomki i don't think you understand what he was saying, and your example is a really small number which he had excluded from his hypothesis.
@@palmomki But they are remainders after dividing by base to some power. 1234 mod 10 is 4, you have your last digit. You divide by ten ignoring the remainder, so now you have 123. 123 mod 10 is 3. Your second to last digit and so on. This is how you can convert numbers to other bases. Pretty simple honestly. As for the second statement. For a random number that statement is true. You have a 1/base chance for each digit to be put in the number. Sure if you pick 111111 it doesnt apply but for big enough random numbers I"m pretty sure it does.
@@palmomki ok, i get what you are saying. But, you get the intuition from what he was saying, right ?
I understand it that way :
Let k be a natural number,
X a random number : X ~ Unif{1,..,10^k-1}
We define Xi such as :
X = X0 + 10*X1 +... + 10^(k)*Xk
Xi ~Unif{0,...9}
let a be a number in {0,..,9}:
Frequency of a in X in mean
1/k*E[sum(1[Xi=a])] = 1/k *k*1/10=1/10
And this holds for any k, so
The frequency of a digit appearing in a whole random number (defined above) is 1/10
(Generalization give 1/b)
@@jujumw5918 But are primes random?
Some people talk with their hands; James talks with his head.
He reminds me of Sir David Attenborough.
@@deplorableneanderthal1265 Sir Attenbobble?
Huh yeaa
Perhaps I had 1too many glasses of wine (4)... but for the first time of my life I got motion sickness from watching someone bob their head.
They were wrong. You can't have a prime number only made of the digits 7 (exept for the prime 7). Because:
77 divides into 11
777 divises into 111
7777 divides into 1111
And so on. A number made of only the digit 7 would be able to be divided into 7*111111111...
So you cant have a prime number only made of 7's (except for the prime 7 of course).
This means you:
- can't have a prime only made of 0's.
- could have a prime only made of 1's (as far as I am concerned)
- can't have a prime only made of 2's and/or 4's and/or 6's and/or 8's because they would divide into 2. (except for the prime number 2)
- can't have a prime only made of 3's and/or 9's beacuse it would divide into 3 (except for the prime number 3)
- can't have a prime only made of 7's. Because of the proof over.
Conclusion: If you want a prime number only made of 1 type of digit, the digit must be 1. (Excluding the primes 2, 3, 5 and 7).
I hear 'all 7s' and immediately go 7 * 111111...111 etc.
*Edit for a few people:* I'm not saying anything negative about anyone in this video, just bringing to light an error that was made for viewers. I understand completely that it was an on-the-spot discussion and that errors can and will be made, and in no way was I trying to be disparaging.
I didn't in any way expect this to get as many likes as it did, so thanks, I guess?
Got eem
yea, they didn't really think about it on the spot, if given a few seconds they would have probably realized.
Proof that not always the brightest of the minds can detect the obvious
Maynard was focused on the main video explanation, obviously he knows that.
@@randomdude9135 Brightest minds will always detect more obvious stuff than others in the long term.
I really liked the exposition at the beginning! It helped put this whole thing into perspective.
For a guy who won a Fields medal he seems remarkably relatable and down to earth. That combination of intelligence and ability to communicate his work is admirable
I like the little prelude at the beginning, it's nice to see style changes every now and then.
He's back!
This guy is an actual legend.
Yes, he is actually a legend
@@akshaj7011 let's hope no one likes my comment until yours gets 7 likes.
@@hewhomustnotbenamed5912 His comment actually has 7 likes now. But I can't possibly deduce if that happened before your original comment exceeded 77 likes or not.
@@lonestarr1490 I can but don't want to.
The wayback machine is an online archive of millions of internet pages at different times, but I'm too lazy to check this TH-cam video at different times.
You could try it if you want.
The very end made me smile, when he was talking about the random number he gives when asked. I'll have to do that myself from now on.
Seeing a number that large printed out like that gave me goosebumps. It's obvious but at the same time absolutely mind blowing to see it like that. It's like staring into the abyss.
What is wild to me is that there are infinitely many primes larger than that prime.
Hey mate, congrats on the Fields Medal!
7 is the only prime with only 7’s bc all other ones will be divisible by 7
Or 11 or 111...
@@maximilianlorosch936 Assuming 111 had no other factors, it doesn't follow the same pattern as 7 because 1 isn't a prime factor. And if you take 11, then it still doesn't work.
@@underslash898 no he means 77 or 7777 or 7777777 is divisable by 7 OR 11, 111, 1111 etc
@@wierdalien1 Ah, that makes sense
The same can be said with 2, 3, and 5.
They say Matt Gray is the bounciest man on the Internet but James could give him a run for his money!
Numberphile videos, I always get lost almost immediately, but nonetheless find them utterly compelling from start to finish.
Heartiest congratulations to James on his Fields Medal 2022
7:50 A number consisting of all 1's is a "repunit" (1 == unit), but with all 7's would be a slightly more generalized "repdigit". BTW, repunits or repdigits can be specified for any base. For example Mersenne numbers are base-2 repunits.
Congratulations James on your Fields medal!
Gotta love how he used the non-number "gazillion" in this video!!
How many south americans does it take to change a lightbulb?
A brazillion.
I had never thought that I will ever see an interview of James Maynard. So happy
JamesMaynard seems like he is rapping ,the way he is enjoying while delivering the whole idea, maths seems to be like music😍😍😍
Congratulations Doctor Maynard!! 🎊)
Ok, but here's an idea: if we can similarly prove that there are infinitely many primes whose binary expansion has no zero, that would mean there are infinite primes as strings of ones in binary, which are always 1 less than a power of two, which are Mersenne numbers, which are linked to the perfect numbers... So it would function as a proof of infinite perfect numbers!
Coool, even though idk what a perfrct number is
Is there already a proof for infinite mersene primes? Bc maybe that part is already proved
Unfortunately I think the proof that there are infinitely many mersenne primes is still unsolved meaning it's probably harder than this
guys plsss stop pointing out the same mistake of the 77....77 being divisible by 7 and 11....11. It's already pointed out so many times that I cannot enjoy reading the comment section
James Maynard, lead singer of the band LOOT.
I'd listen to LOOT
Great intro Brady -- and great video as always
Are there infinite number of primes with their all digits being prime?
Do you mean just those containing 3, 5 or 7, or do you consider 1 to also be prime?
Probably. Not proven though
@@carltonleboss You forgot 2 😜
@@markzero8291 oh yeah
@@carltonleboss 1 is certainly not a prime number.
Here after he won the Field’s medal
0:47 That seems like a ridiculously well distributed set of digits!
that is the expected distribution at that many digits
Wild guess it has something with “1” being at the end (as it is kinda second least divisible digit when being with other digits before him) after 7
There was an earlier video on the channel talking about something similar, where as you approach infinity, all digits exist in every number
Also, why did these guys agree that there might be infinitely many primes that are made up of all 7's? Wouldn't it be divisible by 7???
7s*
Indeed, the only digit which could work is 1.
Imagine switching one of the pages and the number is no longer prime
James: explains his mathematical proof.
Me: wow those books look like glasses full of beer.
This prime number is without the digit 7 if you write it in base 7! (Or base any whole number between 1 and 7, inclusive.
7 is the first weird prime. Composite numbers don’t feel random, 2 and 3 are subitizable, and we have 5 digits on each hand and foot. It’s also the first harmonic ratio factor that starts to sound a bit alien. It’s also kinda near to 10/(the golden ratio), while still sharing no factors with 10-intuitively, the significance of this maps to the “random walk” feel of the units digit of every multiple of 7 (in order).
If you take the latest current known prime, and convert it to binary, you end up with a number with only one and zeroes, that will be at least three times as many digits. Another way to do it, would be to convert the prime into the base of itself. That would be 10, which also does not contain 7.
At least bases are mentioned.
My Proof: Take any prime numer P.
Now represent P in Base 7.
@@garyz2674 I feel like you don't quite get how bases work. 7 in base 7 is represented as the digits 1 and 0. That doesn't mean it's now the number three higher than 7 and has new factors.
Who are you so wise in the ways of Science?
Congratulations Professor Maynard! I'm sure your work will contribute to more discoveries in the future.
How easy would this extend to "primes that have no 41, but can have 4 and 1"? (Well done James!)
Like 491?
James Shaking his head "infinite" times.......
He's gonna do a supertask.
Congrats on the fields medal, Dr. Maynard!
All numbers containing only 7s in them are divisible by 7, so other than 7 itself, all those numbers would be composite numbers.
No one cares that we're early guys. No need to spam the comments with it.
*passive spamming
Matt Parker would definitely print that number. Well done, Matt 👍
Video uploaded 7 minutes ago
Coincidence...
I think not
and uploaded on 20/11/2019, 2+0+1+1+2+0+1+9=16
1+6=7, definitely not a coincidence
And had 7 likes,when i commented
If you write all the primes in base 7, there is no prime with a seven in it.
In fact in that case there isn't even a number called 7.
@ Sure there is. "10" is called seven.
@@OMGclueless In base 7 there sure is a 10, but the symbol 7 is absent, in all otherworldliness. To rephrase my point: would you count something that, by definition, you say does not exist in the context you've postulated?
@ The number seven exists in base seven and it's called "seven", just like the number ten exists in base ten and it's called "ten".
What doesn't exist is a digit "7" to denote seven, the number exists just fine.
YT recommend this video after he won fields madel
7 is chosen most often because people see it is the least ordinary number.
Remove the evens -> 1,3,5,7,9
Remove the first and last -> 3,5,7
Remove the middle -> 3,7
Pick the second one remaining -> 7.
Oddly enough - the 7 of diamonds is a common card people choose when thinking of a card.
3:20 Just curious: does proving that the sum of 1/n for every positive integer n with no 7 in its decimal expansion involve a lot of the Inclusion-Exclusion Principle?
I an fascinated by primes and all the videos from Numberphile that feature primes and James Maynard. I'm also fascinated by interesting facts about primes and the Twin Prime Conjecture in particular. For example you know the Pythagorean Triplets like 3, 4, 5 where there are right triangles with all integer length sides, right? Well, every pair of twin primes is part of an integer solution to a^2 + b^2 = c^2 due to the fact that (n + 1)*(n - 1) always equals a perfect square, so multiplying any Twin Prime pair produces a perfect square and you have a Pythagorean triple on your hands. I think a geometric type proof combined with using imaginary numbers could potentially prove the Twin Prime conjecture. After all, if we ran out of twin primes we'd run out of this class of integer solution to the Pythagorean theorem and we'd run of of rational fractions to fill in the spaces between all those irrational numbers on the number line to express completely unique slopes as angles get smaller if it wasn't true. There can always be a prime just below the square root of a perfect square and there can always be one just above the square root of perfect square. We just need to prove there are infinitely many perfect squares where primes exist in both places. Why not? What would stop them? So this seems "doable" to me. (I wonder if James is reading this...) I also have an idea that primes come in families that form a series where if you only take the primes in a given family, they plot nicely in a graph and the higher up the number line you go, the first element of new families keep appearing. (KInd of a recursive, fractal-like way of generating more primes to fill gaps as needed). That's just a wild hunch and could be baseless, but facts like if a member of the tribonacci series is a prime, the index of that member is always prime lead me to such conjectures. For example, the primes that appear in this series form a sort of family and the indexes that match up to the primes also form a sort of family. Both are subsets of all primes, but what's the underlying connection? If the index is prime, there's no guarantee the member is, but if the member is, the index must be. It's so interesting to me. Why should such a fact even be true? I'd love to talk to James if he ever had some free time to do it..I'm a total prime number nerd and love them.
prime numbers are so fascinating
Wonder what's on the brown paper that is framed on the wall behind him?
I was expecting "Dr. James Maynard, maths professor at Oxford" to be an old, distinguished, bespectacled, balding, grey-bearded cat. Turns out he's a KID!
a number with all 7s is not prime, because is divisible by 7
It took me until the end of the video to put my finger on it: James has the speech cadence of Christopher Walken.
I assume once you get to sufficiently large #'s of digits, each digit has a roughly equal distribution of being any number overall (when averaged out over the whole length of the number) Your example of the 3 volume number follows this roughly even. Thus, the likelyhood of the number having no 7's is basically just 1/10 per digit, and then multiplied to the exponent of however many digits there is in length, and this gives the rough distribution of primes at that digit length that we would expect to contain no 7's, (or any other chosen number
It’s crazy to think that this number, being over 10 million digits long, is merely somewhere between G1 and G2 of Grahams Number (G64)… G1 being 3↑↑↑↑3 (which is equal to about 10^107,000,000,000,000) and G2 being 3↑↑↑↑3 to the power of 10^107,000,000,000,000). Now repeat this hyper-exponentially only 62 more times and finally you’re at Grahams number!!!!
Wrong, G1 is 3^^^^3. Just 3^^^3 is already far too big to fit in the universe, as 3^^^3 evaluates to a power tower of 3s that's 3^27 high. You'd have to write the power tower of 3s from here to the sun, and then evaluate it from top to bottom. G1, or 3^^^^3 is 3^^^3^^^3, so you then have to evaluate 3^^^3 layers of power towers which are themselves power towers. G1 is vastly bigger than you think.
So, what about prime numbers constructed wholly of prime quantities of prime digits?
12:04 perhaps it's because 7 is the largest single digit prime number
8:07 not 7s, they would be divisble by 7
A string of 7s is just the same string of 1s times 7
Wait literally the *only* thing I care about is Numberphile videos with James Maynard. What the hell, TH-cam. Were you ever going to tell me about this?
I love watching videos of Bouncy Dude. You can tell the wheels are spinning. Or genius pistons maybe...I don't know. Whatever..... when you see it, things are fix'n to get solved!
0:47 Maybe someone could find out what is prime number bigger than this one containing equal number of each base 10 digit.
Dr. James Maynard is on a first name basis with all the primes
All sevens: it is divisible by 7 and the number made all of 1s with the same number of digits, so definitely not prime! Only works with only 1s to get (possibly) a prime.
I'm trying to think of a reason why you are wrong and getting nowhere. Well spotted, Vincent.
@@smrusselkabirroomey7396 It is easy to miss that when you get asked it on the spot. I know I have done it many times, thinking about something later and realizing I missed the obvious! Prof. Maynard had a lot to think about, making sure he got all the bits he wanted to talk about, in a decent order, clear and concise, etc.
I wonder if there's an infinite number of primes made up of repeating 1s. (Or, to put it more rigorously, are there an infinite number of primes that can be defined as the sum from 0 to n of 10^n?)
edit: im dum dum who didnt watch the video through
I think he meant to say a number with only 1 and 7
@@ducktectivewhitewings9276 7:47
I'm really glad you showed the clip at the start of the largest known prime and how evenly the digits are distrubuted within it. Really puts into perspective how uncommon a prime with absolutley zero 7s in it would be, and yet there are still infinitley many of them.
The somewhat hard proof might mean that its not so obvious this should be the case. If the probability goes to down to infinity to get a prime number with some property, but you have infinite "random" numbers still to go, is it guaranteed you will always have infinite numbers such as these? Maybe Trump knows
absolutely*
Take any number, and remove all the 7`s. You will get a new number without any 7`s 😆
I'm surprised that anyone bothered to find the digits and count occurrences of each digit-value. And print and bind the thing!
@@rosiefay7283 that' what computers are for
For a serious mathematician, i like that this guys always got a cheeky smile hiding
he always knows something you dont :P
It´s not a cheeky smile. It´s a lack of conversation skill. He is very insecure. You even see it off camera. But he is a cool dude.
@@michaelhendriks9006 Doesn't sound insecure to me
@@ihsahnakerfeldt9280 His body is dancing while talking.
@@azap12 So? How does that show he's insecure?
Hey, he's one of the solvers of Duffin- Schaeffer Conjecture.. crazy smart dude
Professor's looking like he's really fascinated by his discovery. He can't sit on his chair calmly 😊
I looks like he's a marionette controlled by a puppet master who bounces his puppet to show that it's speaking.
hes dancing
heads be bopping
Because he's very buoyant about his discovery.
How still would you be sitting if you were being interviewed about something meaningful that you had discovered?
Isn't a number with all 7s divisible by 7?
7
Kartik Nair yes. I think so.
Kartik Nair 7=7*1
77=7•11
777=7•111
7777=7•1111
77....7=7•11....1
oh, that's true, i didn't notice until i saw your comment xD
I think you're tight. I wish I'd spotted that.
Who's back to this after he won the fields medal?
Legend has it that James is still shaking his head.
He sure got the groove! B)
I feel discomfort when watching him move this way
@@sebbe4717 I usually watch at 1.25x but it was too shakey
Bobbing to the beat of a different drummer 👍
??.
I see the title and i immediatly think "ah 13 right?"
Whenever primes are involved, mathematicians go ever so slightly bonkers.
That's because primes are like glances at the base code of the universe.
Xario Withoutalastname how so? At its root level, why is a number that isn’t divisible by any other # special?
@@codycast Not the single primes themselves, but the complete set. It is completely deterministic but has MOST of the characteristics of a random distribution. Most of the he few (non-trivial) patterns that we now of, are still a mystery to us. In the vid, it was shown that the sum of the inverse prime numbers diverges, but JUST barely. Prime factorization is the base for contemporary cryptography, the Zeta function, which is basically prime factorization in the complex plane, contains one of the biggest unsolved problems in current math. Primes pop up in every area of math and are so fundamental that even natural evolution has stumbled upon them several times as a solutions to different problems.
Xario Withoutalastname fair enough. I guess I just don’t know enough to have a proper appreciation.
I wonder why the video didn’t show the largest prime number known without a 7
@@codycast Probably because it's not very large and thus not very impressive.
I think there was a mistake here. The only repdigit prime possible is all 1s, repunit numbers. A repdigit number with all 7s is always divisible by 7. Still, proving there is infinitely many repunit primes would be super cool.
WHOS HERE AFTER HE GOT FIELDS MEDAL?
Congratulations for winning the fields medal! You certainly deserved it!
I wanna listen to the imaginary disco music that he is jamming to
2022 Fields medalist!!
Simple: just find a prime in binary. No sevens
He stresses it being in decimal.
Jerry Rupprecht actually base 2 to 6 work because they don’t have a 7
base 7 has no 7
You can redefine the base 10 numbers so that 7 doesnt exist anymore
@MATTHEW GOH CHIN LIN (Student) it's not a whoosh stop using that at every possible opportunity
"They disproportionately choose 37."
In a row? Hey, try not to choose any two-digit numbers on your way out to the parking lot!
6:21 - That's the first question I wanted to ask in the comments! You guys are amazing!
This maths dude be trippin'. My man can't keep his head from bobbin'.
Hey James, congratulations for winning the Fields Medal for 2022 for your contributions to Number Theory! I was wondering why old Numberphile videos that I've already watched are showing up in my TH-cam feed all over again ... and now I know! Nevertheless, I'm gonna re-watch them all over again.
James is exploring what he loves, on the frontier of human knowledge, with such humour and enthusiasm, and who knows where this research or the techniques being developed could lead. Great video, thanks !!
Wow, keep your shirt on.
Pioneering mathematical discoveries are often attributed to the courage and inventiveness of youth, James Maynard we salute you!
7:40 all 7s doesnt work becaus a number made up of n 7s will always be divisible by n 1s
7777.... will always be divisible by 7
You'd think that would already be considered, and dismissed when you're already having the 1111.... issue that you've considered.
A number of any length will all 7s will always be divisible by 7
Fast forward in 2022, James Maynard WON the 2022 Fields Medal
You can't have all 7s because it's divisible by 7; specifically a number that's 7 repeated n times is equal to 7 * 1 repeated n times (responding to the discussion @ 7:30 )
5:06 Holy moly a quadruple integral! *Needs a lie down in a quiet room*
So what you're saying is that the treasure was the techniques we made along the way?
He's going to be lecturing the first years linear algebra 2 next term, I'm pretty jealous
Lucky!
Try just going to lecture anyways.
rahowhero X yeh I could do, there’s nothing stopping me. You can just walk into the maths institute and into a lecture, don’t have a register and you don’t need to scan your card at the door or anything.
@@henryginn7490 lol. You dont at any uni where I live, nor uk or oz. Usa?
rahowhero X James Maynard is at Oxford which is in the UK
All single digit numbers can be divided by that digit. One way to see this is consider they all can be multiplied by 10, or 100, or 1,000, etc. Numbers composed of only the digit 1 can potentially be prime because dividing by the digit in question (1) doesn’t count.
Here comes the Fields Medalist
Can’t have primes with all 7’s lol. They will always be divisible by 7.
U cant have a prime no with all 7 beacuse the no. Will be divisible by 7.
1777777... - this is they thought about.
well 7 is prime and all 7's
@@soulstealingginger3612 oo
7:00 As for the repunit 1, it is not only that multiples of 3 don't go.
It can only be a prime if its number of digits is a prime.
For example if n=(10^14-1)/9=11111111111111 consisting of 14 digits, you can write n=11×1010101010101=1111111×100000001=11111111111111, because 14=2×7.
idk if i misunderstood this, but he also said that only 1 and 7 are possible to use for repunit primes, but won't any number with all 7s be divisible by 7? or do you exclude the number itself when considering whether its a repunit prime