Large Gaps between Primes - Numberphile
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- Опубликовано: 27 сен 2024
- James Maynard on discoveries about large gaps between prime numbers.
More links & stuff in full description below ↓↓↓
More Maynard videos: bit.ly/JamesMay...
Prime Playlist: bit.ly/primevids
The 20 August paper: arxiv.org/abs/...
The 21 August paper: arxiv.org/abs/...
Terry Tao interview: • The World's Best Mathe...
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A little extra snippet on just how much Dr Maynard loves prime numbers!!!
ruclips.net/video/muVcPi7oWWY/видео.html
Numberphile can we have Complex number next time please
Stephen Su do you even math
Numberphile love the fact that we are only inhibited by our patience.
I don't always understand or keep up with these videos, but I've still learned a thing or two and love watching them.
+Numberphile can you request Andrew Wiles or atleast reach out to him and make a video with him?
The fact that maynard independently proved this conjecture within 1 day of Tao's collaborative effort is astounding. This kid is wicked smart.
Add to this that the "competing" group was "barely extending" an already fertile ground (not to say it was easy...).
While Maynard did a more disruptive, or creative approach.
I am pretty sure he did not complete the proof in 1 day. He surely published it within 1 day.
It's amazing how often this happens - people working concurrently and independently achieving the same result at about the same time. Be interesting to know how the results came to be published just one day apart though - presumably the first result precipitated the publication of the second.
There should be a theorem about the gaps between two consecutive papers on gaps between two consecutive primes.
guys! humanity must find the last prime number!
2X3X4.........X infinity + 2
2X3X4.........X infinity + 3
2X3X4.........X infinity + 4
2X3X4.........X infinity* + infinity* must be equal
So there is an end of prime numbers
"Terry Tao only beat me by one day."
That's pretty badass, dude.
The US really needs 53 states, then we really could be "One Nation, Indivisible...."
Puerto Rico, Guam, Samoa... we've got some candidates already!
Or we could just throw out all but the original 13.
DC
James Flaum
just 51 actually
51 = 17 x 3
All prime numbers except 2 are odd, this makes 2 the oddest prime.
Even though it isn't odd?
Dan Peal because it is odd, as in not like the rest
*facepalm* you missed my pun, friend
How's that for irony?
Wouldn't 2 then be the _least_ oddest prime?
What sound does a drowning mathematician make?
loglogloglogloglog
Shane Dobkins Specifically, a number theorist ;P
Shane Dobkins Terence Tao!:D
lol
Are you THREATENING me?
++++
"times a small constant c"
*writes a tiny letter c*
Anyone else think James Maynard would be the best math teacher ever? He's so polite and enthusiastic
superpanda9810 I wish he was mine.
_He’s all mine._
Dr Maynard talks about math with the kind of genuine excitement only a child would show, I loved every second of this video!
Where's Ramanujan when you really need him?!
Here!
Srinivasa Ramanujan : My heart just skipped -1/12 beats
That "here!" thing , was so unexpected!
Where's EULER?
I laughed aloud.
I just want to thank you guys for continuing to bring cutting edge maths into the public eye.
naughty brady using comic sans.
Still better than Papyrus
+
I wonder if Matt gave this a go, and got it almost right...
I can see that you gave something a go too. *pats in the back.*
SoyLuciano Someday he will discover the Parker gap, a gap that's correct except for infinitely many exceptions
The Parker Primes?
key word "almost"
What will happen if we find a sequence that decrease the gap between prime numbers?
Dr Maynard has won the Fields Medal! Congratulations!
coming in second to a team of four people including Terrence Tao is really impressive
These videos with Dr. Maynard are great!
Congrats on the Fields Medal, James !!
They split the 10.000. But they split 5000 to each response or 2000 to each person? I think James should get 5000 since he did his own work by himself
Lucas Aielo I was thinking the same thing... I'm guessing they split 5 way equally.
Imo they should split 1k between them and give 9k to me.
they're professional mathematicians, of course they'll be calculative and come up with some number theorem to split it equitably.
Well, of course, what you want here, is a geometric compromise between those two most obvious strategies; so that ⅓ goes to James, and ⅔ goes to the 4-way collaboration, to be split into 4 equal, ⅙ shares.
That way, the quartet gets twice what the soloist gets, while each member of the quartet gets half what the soloist gets.
As for the odd 4¢ (6 · $1666.66 = $10k - $.04), they'll have to fight amongst themselves over that, just being thankful that the quartet wasn't a trio for purposes of this rule.
And before you ask (if you even thought you had to), yes, I *am* a mathematician.
1*4000+4*1500
Maybe?
I went on internet to rest from math, but looks like I won't :D
Kakvo imam zapažanje, poznata mi je ova slika skroz. Pozdrav brate balkanski :D
Pozdrav zemljače :D
Crazy drummer
Lol i do the opposite
That factorial proof is so simple yet cool
Judging by that Rolex, Dr. Maynard, I think I know where the 2000 bucks went
Not a Rolex. Looks like an Armani.
Congrats James!!!
I really love your channel. I'm a Biochemist, and most of my life math was just a useful annoyance I had to study for 2 years. I've enjoyed watching your channel so much more than I though I would, and it;s given me a whole new perspective on the meaning of mathematics. Thank-you for doing this.
My tiny improvement: instead of n! you just need the product of primes
What about 2*3*5*7*... + 4? ;)
you forgot to account for prime powers ;D
@@user-me7hx8zf9y not following... 2*3*5*7...+4 is divisible by 2 and therefore not prime.
@@Quantris Yeah I scrolled up to correct myself
@@Quantris 5 am number theory gang wya
I find it absolutely fascinating how Maynard and the other group had completely different approaches to the problem, but got _the exact same_ formula for large prime gaps.
Is there some strange connection here? Or was that formula already hypothesised to be the solution, and they simply used different approaches to proving it?
Great question
Why is it that prime numbers, constants and their relations and patterns are so intriguing?
I haven't even studied math, had OKish grades in school, but now that i am free of the constraints of school or using math at work it all starts to have such a fascinating glimmer to it.
It all started with SDRs and i was fascinated how, with help of i.e. the fourier transformation, you'd be able to extract signal from noise that no human ear could even guess they were there. And you know if you say Fourier, you say "e", "pi", "i"...
That was where my jouney began.
I don't know, but i am in high school and these prime related videos are particularly interesting to me
Me pranking a high schooler: “Find a prime number larger than infinity factorial.” 😂
Being a mathematician might just be the best job in the world, seeing as how you get addicted to your job... No wonder all of these guys smile all the time :D
last time I saw so many logs in one place, they were building a cabin!
Or a really huge bonfire!
Or an equation about prime gaps!
**slaps knee**
*thumbs hooked in and pulling out maths braces*
bonfire(x) = log(log(log(log(log(log(log(log(x))))))))
New Fields dropped
You can use Bertrand to show that if x
Uploaded 8 minutes ago; video is 9:26 in length; 116 likes... You people have good faith!
Now that I have finished the video I can confirm that your faith was well founded.
But the last minute can't ruin the whole video, can it?
You're new here aren't you
You are not watching the videos in higher speed?...
2 years and 1 month past only 3 dislike has increased so we can conclude they were from rival channel :D
Would realy like to see more of Dr Maynard!!!!!!!!!!!!!!
That is a really cool expression.
Mind blowing that one can discover and tinker with something like that.
Nicely done!
Using the simpler expression and simple minded solution, if you wanted to look for an arbitrarily large prime, you could start with X! and then work downward (X!-1) to avoid the known gap.
congratulations for winning the fields medal
0:20 - "... a very high-school argument."
I went to Cambridge to do maths, and I didn't see this until my first week there.
I think he means that an argument a high schooler could understand since in high school you could understand factorials and adding them is not a prime
I did just realize that there’s a space of x!-x!/x between every off limit section, which means the gap in which it’s unpredictable grows by a factorial too
I've always been fascinated by the twin prime gaps of the same size such as 199 to 211 and then 211 to 223. Prime number 211 has a gap of 12 in each direction. I wonder if this can be done for every even number.
So, the purpose of using logarithms are to deconstruct a variable exponent, right?
What was the original equation that required all those logs? That's the link or video I want, Brady!
Maybe there are prime gaps between p_n and p_{n+1} of length c^{sqrt(ln(n))} for some c>1 or better infinitely often.
Hey, don't be so rough on your formula! logs of logs can take time to calculate, but they make the large numbers significantly smaller! I think it's a great and efficient formula :)
It's the first time I've seen James Maynard on a Numberphile video. I look forward to more.
Speaking as one of your innumerate viewers, I'd say good job, nice delivery and he look as if he doesn't get out much, like a proper mathematician. Appearances can be deceiving of course.
Please do a video about Maryam Mirzakhani and her work on geometry
Terry Tao & collabs straight up ninja'd James Maynard
What you think of the graph of `f(x) = prime(x)/log(prime(x))/log(x)` ?
I like how his 0s look like hearts
Is there any proof for the longest same number gap between consecutive primes? Eg if there was 6 spaces between two primes, then 6 spaces to the next one, then again and again?
The most that consecutive six would get is 4
Interessting how +6 was used there.
I once played around with C a bit and found that the gaps between 2 primes "tend" to be multiples of 6.
With "tends" I mean:
If you plot the number of primes with gap x vs x you get something similar to a saw.
dropping, dropping, oh: X is a multiple of 6: increase a bit again, dropping, dropping, oh: multiple 6: increase etc.
All primes can be expressed as 6N+1 and 6N-1
If the prime is 1 less than a multiple of 6, the gap to the next prime is either a multiple of 6 or 2 more than a multiple of 6. Assuming the gaps are reasonably random, each case should account for about half.
If the prime is 1 more than a multiple of 6, the gap to the next prime is either a multiple of 6 or 4 more than a multiple of 6. Again,
each case should account for about half.
So, the gap between consecutive primes should be a multiple of 6 about half the time. For smaller primes, you can expect this to be a little less than half because the non-multiples of 6 get the first shot, but the ratio should approach 1/2 for larger primes.
SAW MATT PARKER AFTER SCHOOL TODAY AT THE LATYMER SCHOOL, EDMONTON!
8:16 Hmm, I think James Maynard forgot to square the denominator in this video so the formula differs from both his and the other team's papers.
A gap of n-1 primes appears much earlier than n!+2. There will also be a gap starting at the least common multiple of 2 through n, plus two, a much smaller number than n!+2.
I dunno how to express this but...
If you take a set of 10, and reduce it to the digits that could possibly be prime (numbers ending with digits 1, 3, 7, and 9). This set of 4 numbers will always be divisible by primes or exponents of primes if they're not prime. I'm not sure if this is known but I assume that someone else must have stumbled across this... but once you know this it seems pretty obvious why there are gaps and how big they should be... they're predictable because they grow as the pool of primes and prime exponents grow and the size of gaps should be fairly easy to figure out from that... likewise where the primes should turn up because it's simply where the multiplies primes and prime exponents don't line up >.>
Though I haven't really spent much time on primes and I'm not really good at math, so I think this should be well known but it seems that people don't know it or it just isn't significant...
8:05 loggers logchamp
1:48 I see that long-scale billion there.
Love the way he explain it
Instead of n! , we may use p! (Defined as all the primes less or eqial to n)
So we're using "billion" from the long system then? ;)
Surprised nobody immediately pointed out its unnecessary to multiply by all 100 numbers but only take the product of primes
very inspiring..
Just a point of reference. It's pronounced 'Air Dish' I only know this from reading the book 'the man who only loved numbers' a fantastic read by the way.
Please make a video showing how this proof on prime gaps is related to Yitang Zhang's work on prime gaps.
this might be a weird question but are the smallest gaps between the primes equal in size every time between them
is that how mathematicians talk about currency? "Numerical bits of money"
the twins exist forever since such gap is seeded in the first two prime (obviously odd) numbers 1 and 3
So is that statement at the end saying that the largest gap is less than that expression?
There should be video on Numberphile about harmonic analysis, Fourier series or wavelets. I mean, isn't it insteresting that we can build almost every function from many tiny parts? Thumbs up if you are also interested!
I controlled myself by not commenting before watching the full video
Who worked longer on his paper? Can you quantify that at all, do you log your time on a certain topic?
What can we say about n!+1 ? Clearly the previous argument tells us nothing because every number is divisible by 1 so I guess they can be both prime or composite depending on values of n: 3!+1=7 4!+1=25=5x5 but can we say something more precise about the form which "n" has to be to make the expression prime? for example 3 is odd however 5!+1=121=11x11 ...
ZalgoOfficial Google "factorial prime" also look up sequences A002981 and A002982 in OEIS.
Consecutive days where solutions are found to a problem relating to consecutive prime numbers are often far apart.
Whose work was more accurate? Or did both groups end up with the same formula?
*Likes video before watching it*
Zero Cool I don't know why I laughed at your comment.....
Likes comment before reading it
Timothy Smith I thumb them because I like the channel.
This makes me wonder what the ratio of primes/whole numbers is and how that changes over regular intervals in magnitude on the number line.
The biggest difference between the prime numbers is a better discription for the trigonometry. The biggest gaps between the prime numbers are the functions of tangent where the prime number "x" is to a distance of tag (lnx ^ (1/5) * and * i) of the prime number "y". The angle must be between 0 degrees up to 90 degrees, this due to the calculation, in other words, to minimize the effects of the inflections of the function. When a prime number approaches 90 degrees or zero degree, we have the biggest difference between the prime number. A fact is snoopers that as in a circular area between 0 and 180 degrees we can only build triangulos rectangles when two of his extermidades are between 0 and 180 degrees, to functions it expresses tangents exatclly the relations between prime numbers distances. In an angle from zero to 90 degrees in a trigonometric proper criculo the characteristic of the tangent, of tendency the asymmetry is easier to calculate the decimal number that is the biggest between prime numbers. I have not computational power but of the house 43 to 87 of the number pi, tranformation this number in decimal, this is the biggest distance between a prime and different number. I believe that we will find the first one in the order of 10^123 prime numbers. Interesting, since for convencion the 1 is not a prime but others, 2 and 3, are the first two prime numbers. It is important to notice that the function will give an irrational number, which leads to think that between the house 43 and 87 of the number pi we add the first term of the decimal house, then the second for the second number I excel and so successively. You can put the decimal number in the channel in order that I know exactly the medium of this distribution? Thank you.
I desperately want to see Maynard and Tao get in a nerd fight.
If you've ever checked out the math collaboration website they post to, it almost seems like they're constantly trying to one-up each other.
TREE(3)! + TREE(3)
That proof skips n!+1, which can be prime. The most trivial example is n=1, which works as 2 is prime. n=2 & n=3 also work, as 3 and 7 are both prime.
I created a program, and the next one that works is n=11 (as 39,916,801 is prime). So there doesn't seem to be any reason why n!+1 can't be prime.
That said, I can't test higher than n=21 in native JavaScript (with my isPrime function). JavaScript integers max out at 2^53-1, and 22! > 2^53.
I guess we just rely on the semi-random distribution of primes to argue that, while n!+1 may be prime, there is an (n+k)! + 1 that is not.
The proof goes
every number from n!+2 up to and including n!+n can’t be prime, hence there are prime gaps of n
The proof specifically excludes n!+1.
Thumps up if you find a partialy erased blackboard irrationally annoying.
Why all the logs? What does it mean to multiply them all together like that?
Why doesn’t this hold for 2!+2 and 3!+3? Why does the argument only hold once you’re factorialising a number where you will have at least one prime in the factorial? Why can’t in general n!+1 be prime notwithstanding first part of comment?
I am a simple man. I see Paul Erdős - I hit "Like".
Why?
@@sillysausage4549 , he is a legend.
Has anybody else found this prime sieve? For a!+b=c, where b is a prime larger than a, what is the smallest b has to be in order for c to be prime? So far, I'm finding surprisingly small bs. The first twenty a,b are 1,2; 2,3; 3,5; 4,5; 5,7; 6,7; 7,11; 8,23; 9,17; 10,11; 11,17; 12,29; 13,67; 14,19; 15,43; 16,23; 17,31; 18,37; 19,89; 20,29. Is this anything?
This is a pretty slowly-growing function. It's only ~390 for x = 10^100. Of course it's a lower bound.
A little better than primorials. The primorial through 251 is just over a googol and yields a minimum gap of 255.
So the length of a prime gap can get as large as you like. A length so close to infinity that there is no practical difference. That's pretty big.
I took Graham's Number * Pi + Kolakoski Sequence - Euler's Constant / Golden Ratio and I got 69 as my answer. :)
He had the board but chose the paper.
Muhammad The Hope
The brown paper is a Numberphile staple :P
ok, there is a question bothering me:
Accepting we are able to proove, that there is an albitrarily large (countably infinite) gap in between at least two primes, does that mean the following for us:
concerning practical matters of reality: will we find a prime, that is "the largest prime", we will ever be capable to find, as the next one is going to be infinitely far away and therefore uncalculatably "far away"? Or is it just one of these "infinity"-paradoxes (what I expect the unsatisfying answer to be), where we will find, that it's just a question of computational power or a question of definiton of infinity?
interesting comment earlier suggests infinity paradox to where the multiplication suggests an infinite number of infinitely large gaps but addition is simply tagged along for the ride... reasonable to expect there is some rendered meaning in that but indecipherable to my armchair and me
MIND THE GAP between the platform and the train!
So, whose paper got the better bound on consecutive primes? Or did they get the same bound in two different ways?
i was just thinking about length between primes!
more specifically if the nth prime over the nth composite converged to a certain point
That would probably grow to infinity, as the fraction of integers that are composite tends to 1 as they get larger.
Considering the idea of X! + Y resulting in an X sized sequence of non-prime numbers, what would be the case if X was equal to infinity?
Very nice!!
So if the gaps between primes can extend to infinity wouldn’t that mean that if there’s an infinite gap between primes there must be a finite number of primes? Just a thought
No, gaps between primes can be arbitrarily large, meaning that if you give me any number, I could find a gap between 2 primes that is larger than that number.
Is the prime gap prove a prove by induction?
BTW, which log is this? Log10 seems an odd number to show up, but you didn't use ln, either.
He Probably means log(e)
Yes, the heat death of the universe is a pretty big problem.
Surprised that Legendre's conjecture wasn't mentioned.
You don't need to do factorial, multiplying the prime number array's elements is enough
Yes, primorials can do it instead.
I still don't know why we need to solve this problem but it's interesting..