Large Gaps between Primes - Numberphile
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- Опубліковано 18 лип 2017
- James Maynard on discoveries about large gaps between prime numbers.
More links & stuff in full description below ↓↓↓
More Maynard videos: bit.ly/JamesMaynard
Prime Playlist: bit.ly/primevids
The 20 August paper: arxiv.org/abs/1408.4505
The 21 August paper: arxiv.org/abs/1408.5110
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!!!
ua-cam.com/video/muVcPi7oWWY/v-deo.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?
"Terry Tao only beat me by one day."
That's pretty badass, dude.
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?
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
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
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!
Dr Maynard has won the Fields Medal! Congratulations!
I just want to thank you guys for continuing to bring cutting edge maths into the public eye.
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?
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.
naughty brady using comic sans.
Still better than Papyrus
+
These videos with Dr. Maynard are great!
coming in second to a team of four people including Terrence Tao is really impressive
Congrats James!!!
"There are a prime gaps bigger than the number of atoms in the universe."
Ultrafinitists:TRIGGERRED
That factorial proof is so simple yet cool
Near the start of this video I thought to myself, "I bet the answer involves a log function...". I had no idea how much I was going to end up laughing at the end.
Congrats on the Fields Medal, James !!
That is a really cool expression.
Mind blowing that one can discover and tinker with something like that.
Nicely done!
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
Big congrats on your fields medal! Well done :-)
Based
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
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 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
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.
Would realy like to see more of Dr Maynard!!!!!!!!!!!!!!
Judging by that Rolex, Dr. Maynard, I think I know where the 2000 bucks went
Not a Rolex. Looks like an Armani.
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
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.
You can use Bertrand to show that if x
No trees were cut down in the making of this video
Mave Flair lumber Jacks (specifically for tutorials)
log
Oops, brown paper o.O
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
Love the way he explain it
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))))))))
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!
Me pranking a high schooler: “Find a prime number larger than infinity factorial.” 😂
idea for a piece of mathy art: you have your x axis be the increment you increase by and your y axis is the prime you start with, and at every point you color it based on a scale from 0 to the highest number of primes in a row included on the graph. i'd be very interested to see how it turned out, but i don't have the brainpower, patience, or resources to do it myself.
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
Please do a video about Maryam Mirzakhani and her work on geometry
very inspiring..
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.
How does the Riemann hypothesis play into the problem of large gaps between primes and do the (nontrivial) zeros of the zeta function tell us something about the gaps between primes?
Please make a video showing how this proof on prime gaps is related to Yitang Zhang's work on prime gaps.
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)` ?
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
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
congratulations for winning the fields medal
Very nice!!
8:05 loggers logchamp
another maynard!!!
8:05 "I'm a lumberjack and I'm OK. I sleep all night and work all day!"
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.
WOW. A new record for Numberphile.
Had me 100% confused within 39 seconds. Well done team !
He had the board but chose the paper.
Muhammad The Hope
The brown paper is a Numberphile staple :P
You have large gaps between primes is because waves expand as they move from the center. And interference patterns have wide gaps.
So, whose paper got the better bound on consecutive primes? Or did they get the same bound in two different ways?
Who worked longer on his paper? Can you quantify that at all, do you log your time on a certain topic?
awesome videos.
Yes !
SAW MATT PARKER AFTER SCHOOL TODAY AT THE LATYMER SCHOOL, EDMONTON!
If the numbers either side of a primorial Pn# are themselves composite, then the sequence ( Pn# - Pn'+1, ... , Pn# + Pn'-1 ) forms a gap of length 2(Pn' + 1) - 1 [ where Pn, Pn' denotes the nth prime and its successor prime, of course ]. The sequence ( N! + 2, ... , N! + N ) 'only' guarantees a length of N - 1.
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 :)
At the beginning of the video that very wonderful proof not only shows that you can find any arbitrarily large gap in the primes, but also can be used as a formula to give the exact location of those gaps, if you were to work out the extremely large value of the factorials. I don't know much about the proof for the arbitrarily long arithmetic sequences of primes, but does that proof use a similar formula that can also pinpoint the exact location of that arbitrarily long sequence? In other words, could that proof be used to easily find extraordinarily large prime numbers?
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...
8:16 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.
god I love this channel
Whose work was more accurate? Or did both groups end up with the same formula?
Looking at numerical experiments, how accurate is the exact bound with respect to what we observe for finite but large x?
Was Erdős's prize money a good predictor of how long it would take to solve the problem and therefore how difficult it actually was (although I realize the larger incentive for supposedly harder problems works against this metric)?
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 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
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.
So if you put in infinity factorial does that mean there is an infinitely large gap at some point? Implying there are a finite amount of primes on the real number line (the last prime before the infinite gap)?
other than encryption, is there a use for finding these large primes?
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
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.
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.
Welcome back James!
More Cliff Stoll, please.
2022 Fields medalist!!
This guy reminds me of fuzzy peaches, sweedish berries, and wine gums for some reason.
For p>9, don't you just have to check numbers ending in 1, 3, 7, and 9, instead of the other 6 numbers in that group of 10 (e.g. 20-29)? That is because numbers ending (0, 2, 4, 6, 8) are not prime because they are divisible by 2, and those ending in 0 or 5 are divisible by 5. While this is still a linear search, it is better than searching all numbers.
Is that 100!+x(in which x=1...100) also the first time since 1...100 where you get 100 consecutive divisors for 100 consecutive numbers?
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
Dr. Maynard, can't the twin prime conjecture be solved by proving Brun's constant irrational?
Can you solve problem 3 on imo 2017?
Just curious
Dr. James Maynard? He solved something that Erdös couldn’t solve, and is quite prolific. I think an IMO problem would be a bit too easy for him.
What's the website he said they publish papers on?
1:48 I see that long-scale billion there.
MIND THE GAP between the platform and the train!
Great video! I've been wondering about the formular Maynard used (n! + n). Intuitively I would say that if we let n approach infinity the gap would become infinitely large, and therefore we would have a point where there was no more prime numbers. In other words; that there are a finite number of primes. Obviously this is not proving that there are a finite number of primes (at all), as Euclid proved /there are an infinite amount/ 2500 years ago, but I would like if someone could clarify to me why this does not work?
My best explanation would be that this gap would come infinitely far out, so there would already be an infinite number of primes, but I'm sure a lot of you have smarter things to say! :D
Just for the record: I know I'm treating infinity in a messy way, and that is probably not a very wise decision.
Hope you can help me :)
isn't what it suggests though is that there are an infinite number of prime numbers that are infinitely distant from each other... i think what helps that "make sense" is that because numbers are infinite there are easily comprehensible an inifnite number of numbers between all numbers and as such no differently for prime numbers... the n! + n formula is just the theorem to describe that principle in terms of primes
"I know I'm treating infinity in a messy way, ..." . . . and that is exactly your problem.
n is *never* actually infinite.
"... if we let n approach infinity the gap would become infinitely large, ..."
... and because n is never actually infinite, the gap never is, either.
"... we would have a point where there was no more prime numbers."
There is no such point.
"My best explanation would be that this gap would come infinitely far out, so there would already be an infinite number of primes, ..."
That's actually the right idea; although once again, not really rigorous.
Fred
because your "n" is predicated to be an arbitrary yet finite natural number. So the statement "for all natural numbers n bigger than two, n! + 1 to n! + n are all composite" does not imply anything about the finiteness of primes, which if you know, are an infinite bunch :p
So is that statement at the end saying that the largest gap is less than that expression?
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.
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.
Where do you get all those logs in number theory?
Its amazing how numberphile surpasses vsauce in the number of ways to blow my mind