Primes without a 7 - Numberphile
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- Опубліковано 7 чер 2024
- James Maynard discusses his proof that infinite primes exist missing each base 10 digit - he uses 7 as his arbitrary example. More links & stuff in full description below ↓↓↓
More videos with James Maynard: bit.ly/JamesMaynard
The paper on primes with restricted digits: arxiv.org/abs/1604.01041
James Maynard on our podcast: • The Badly Behaved Prim...
More Prime Number videos from Numberphile: bit.ly/primevids
World's biggest prime printed on paper: • New World's Biggest Pr...
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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.
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'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
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
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?
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
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).
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?
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?
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 really liked the exposition at the beginning! It helped put this whole thing into perspective.
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 UA-cam video at different times.
You could try it if you want.
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 UA-cam feed all over again ... and now I know! Nevertheless, I'm gonna re-watch them all over again.
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.
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
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.
6:21 - That's the first question I wanted to ask in the comments! You guys are amazing!
Hey mate, congrats on the Fields Medal!
Congratulations for winning the fields medal! You certainly deserved it!
Pioneering mathematical discoveries are often attributed to the courage and inventiveness of youth, James Maynard we salute you!
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.
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.
2022 Fields medalist!!
I like the little prelude at the beginning, it's nice to see style changes every now and then.
I see the title and i immediatly think "ah 13 right?"
Brady is amazing, contagious enthusiasm and genuine curiosity! It makes those videos so fun to watch
WHOS HERE AFTER HE GOT FIELDS MEDAL?
Who's back to this after he won the fields medal?
I had never thought that I will ever see an interview of James Maynard. So happy
Great intro Brady -- and great video as always
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.
I love this guy, more of him please!
They say Matt Gray is the bounciest man on the Internet but James could give him a run for his money!
I loved the prelude/intro - good one Brady!
I wanna listen to the imaginary disco music that he is jamming to
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.
This guy is so sweet. The way he's passionate shows in his body language and tone, makes it really enjoyable to listen to him.
Heartiest congratulations to James on his Fields Medal 2022
Loving the intro to give some extra context. Great addition.
"They disproportionately choose 37."
In a row? Hey, try not to choose any two-digit numbers on your way out to the parking lot!
I kind of loved the postpunk vibe of Brady's apartment and even more the dancy-wavy vibe of James as he talks what he's passionate about.
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.
Nice to see him win fields medal and we need an interview of him.
12:20
~ 7 is salient because of the exotic effect
~ it's the only single digit number that has 2 syllables (excluding glottal stops)
~ 37 is salient because of the law of least effort, because of the recency effect, and because of the exotic effect
~ we generally remember recent (end) objects better than initial (start) or middle objects
~ thus, when giving a list, people are inclined to think of the end object (7) more than the other objects
~ and 3 (thirty) is the least effort initial object to get to the exotic (7) end object
~ on a side note:
~~ “the” is, by far, the most common morpheme and english word
~~ the consonant [θ] “th” requires little effort to produce relative to most other consonants
Fast forward in 2022, James Maynard WON the 2022 Fields Medal
I liked your podcast with James too!
passive spamming**
We want a detailed explanation of Hodge's conjecture, the British Dyer conjecture, and Clay Institute problems
Really liked this video, well done!
Great work. I also really enjoyed his albums 10,000 Days and Lateralus.
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
5:06 Holy moly a quadruple integral! *Needs a lie down in a quiet room*
Wow, so base 10 actually came out of the problem naturally. Wasn't expecting that at all.
Also, I spot Artin's 'Algebra' on the shelf! Great book.
Congratulations Doctor Maynard!! 🎊)
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
JamesMaynard seems like he is rapping ,the way he is enjoying while delivering the whole idea, maths seems to be like music😍😍😍
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?
Regarding the other problems you discussed, what about if the least significant digit is exempt from the restriction? Eg. could there be an infinite number of primes consisting only of 5's except ending on a 3?
This maths dude be trippin'. My man can't keep his head from bobbin'.
A number of any length will all 7s will always be divisible by 7
This is intuitive for sure so I was looking forward to some presentation on how the proof was made. is this coming to the second channel?
A heuristic argument: for given n there are 8 x 9ⁿ⁻¹ n-digit numbers with no 7.
The density of primes amongst n-digit numbers is approx 1/ln(10ⁿ) due to the prime number theorem.
So we would expect to have 8 x 9ⁿ⁻¹/ln(10ⁿ) n-digit primes with no 7. This is far more than 1.
The trouble with this “argument” is that if it was a proof, we would know that there are infinitely many Mersenne primes.
Natural numbers don’t have divisibility properties based on probabilities. Except that they seem to.
This makes me think about the James Grime video "3 is everywhere" I think also on your channel
James Maynard, lead singer of the band LOOT.
I'd listen to LOOT
So what you're saying is that the treasure was the techniques we made along the way?
So cool! Love this channel
numberphile, bringing us fantastic personalities and some neat math too!
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
He's so young to have done something so cool in the field of mathematics.
When you do programming this exact principal applies. You might be making a "fun" project but what you learn along the way can be applied to many fields.
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
i love how every math professor ever has like 30 yellow GTM books lined up on their bookshelves xd
Great talk
A natural way to extend this result: does it still hold if we replace "not containing the digit a0" by "not containing the natural sequence a0a1...ar" (in their decimal expansion)? e.g. not containing the sequence 17?
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.
Infinity in and of itself is quite an interesting concept
And the primes are like a way to "probe" infinity. That might be one of the reasons mathematicians like them so much.
thanks these videos keep me thinking
This video is from 7 months ago, perfect!
7:50 Certainly a prime number cannot consist of only sevens, because such numbers are obviously divisible by seven.
They thought about one and then sevens.
That's wrong. Example: 7.
Stefan Wagner that’s THE example. As in, the only one.
Here after he won the Field’s medal
There was an earlier video on the channel talking about something similar, where as you approach infinity, all digits exist in every number
The happiness when you find a Numberphile video you somehow missed to watch!
What's the biggest prime discovered so far that doesn't have a 7?
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
Does the Erdős-Kac theorem hold for numbers with restricted digits? I don't see how it could. What then would the distribution of numbers by their number of factors be?
Congratulations James on your Fields medal!
....i didnt even realize this was uploaded 24 seconds ago...until i noticed the view count was 0
Next video
Something in math that seems random but actually has a pattern
That has been done. aka Mandelbrot sequence.
@@elbaecc another one
Lol, or just rewatch that video where Neil Sloane holds up graphs that look like things.
6:21 annnnnd Brady strikes again with the question burning in my head!
Congrats on the fields medal, Dr. Maynard!
Getting dizzy looking at him wobbling!
What's the largest known prime with a missing digit?
4621
@@matthewstuckenbruck5834 Not sure if I was clear enough. Perhaps I meant. The largest prime known that doesn't contain all digits.
@@kevsterking it's a joke, I'm not sure that anyone is looking for those. They would be really hard to find
You can start finding some, but they’re pretty common amongst numbers that are easy to work with, and even for numbers that are beyond that. Put “next prime after 111111111111111111111111111111” into wolfram alpha [that’s (10³⁰ -1)/9] and you’ll find that 111111111111111111111191 is prime. This is not surprising due to density of primes: all n-digit numbers whose first n-8 digits are 1 and whose other 8 digits are anything have a missing digit (pigeonhole principle with negative pigeons). Then due to the prime number theorem, for numbers with n digits, the density is approx 1/ln(10ⁿ) = 1/(n ln(10)) = 1/(2.3n). For this to be less than 10⁻⁸ (suggesting none of the numbers 1111...1111abcdfegh are prime) requires n>10⁸/ln(10)=43 million. A similar probability argument would apply if you chose some other set of 8 places to differ from 1 or some digit other than 1 (excluding obvious cases where last digit is 2,4,5,6,8,0). That is, we would expect missing-digit primes to be common enough for primes with 43 million digits.
(For comparison, current largest known prime has 24 million or so digits)
Oh fun - maple has “prevprime” so 7777...7771 (1067 digits) is prime.
One of the great UA-cam channels
12:46 excellent, I like this form of being difficult
James Shaking his head "infinite" times.......
He's gonna do a supertask.
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.
Are there an infinitely many prime with no chains of the same digit? (11,22,33,44...111,222,333. Etc.)
I like the way he casually said notwithstanding like it's a word that's used all the time
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?