Exploring the mysteries of the Prime (gaps!) Line.
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- Опубліковано 5 кві 2021
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Thanks to Kerry D. Wong who originally piqued my interest by sending me this:
www.kerrywong.com/2009/09/06/a...
Here are the papers about finding arbitrarily large gaps between primes:
The first paper: arxiv.org/abs/1408.4505
The paper one day later: arxiv.org/abs/1408.5110
Watch the author of the second paper James Maynard talk about it on Numberphile:
• Large Gaps between Pri...
The 'top dots' are called "Jumping Champions" and you can read more about them being primorials here: mathworld.wolfram.com/Jumping...
Here is loads about prime gaps with references if you want to see the original papers.
sweet.ua.pt/tos/gaps.html
That link was sent to me by my Patreon supporter Bruce Garner (who has also calculated the first trillion prime gaps using C++). That page is excellent; I wish I'd found it before I started writing the video.
CORRECTIONS:
- At 08:07 they should have been p_(n+1) - p_n ≥ 8. Or I guess p_n - p_(n-1) ≥ 8. But not what I put. What I put was wrong.
- Wow. At 15:33 I said "big zero" instead of "big O". I'm so adamant that I say "zero" instead of "oh" for the digit that I finally had a subconscious false-positive!
UPDATE: I actually spotted this one right before release and it looks like responses were pretty split on if I should have fixed that. I didn't. / 1379438185374171137
- At 23:43 I mentioned your ISP knowing what you are searching for but I forgot that https means the rest of the URL is hidden from your ISP. So they would only know you have visited google, but not see anything after the "?" in "www.google.com/search?q=giant...". I should have used a website as the example. I'll do that next time.
- Let me know if you spot any mistakes.
Thanks to my Patreons who are vital in keeping the videos coming. Future Matt has a very expensive rider.
/ standupmaths
As always: thanks to Jane Street who support my channel. They're amazing.
www.janestreet.com/
Filming editing by Alex Genn-Bash
Graphics by James Arthur and Matt Parker
Music by Howard Carter
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/collections/b... - Розваги
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Two things! Use this link to get 78% off Private Internet Access and three months free: www.privateinternetaccess.com/standupmaths
Secondly: I've put the 'behind the scenes' on how the number line was made on Patreon. Spoiler: it involves a spreadsheet. www.patreon.com/posts/49699535
Love PIA! Learned about it from LinusTechTips and been using it since for years!
Mat. Check out my math vid's i made. Of a program that no one has made. Please. I am sick and i may die. I don't know yet as i have not gotten tested yet. But i will. Talk to me i want to give them to you as tools for teaching. Freely.
You do realize you said "because they're all odd numbers" when referring to the first one million primes, right?
Hey Matt, I spotted a small mistake for the corrections list. At 5:11, the GAP axis shows the numbers 0-16 which should be 0-160.
PIA has been bought by Kape Technologies(formerly crossrider), since that time in court proving they didn't log. OVPN is currently the only proven non-logger from a court-case that is still the same company. Other VPN's are unproven (PIA is among them now, read into Kape Technologies and their crossrider days making malware and adware) and NordVPN had a data breach and didn't inform their customers that they might have been leaking their data untill a year after. With everything online, a small provider might have sub-par security, but they are also a smaller target but it's always a risk.
VPN's are not a risk free privacy guarantee. Not only have huge companies suffered data breaches, but the "hiding from your ISP" argument is *ONLY* valid if you trust your VPN provider more than your ISP.
Poor Past Matt, always getting interrupted by that know-it-all from the slightly less distant past.
Story of my life.
You thought the Parker square was named after Matt Parker. Actually, Matt himself is merely the human example of the Parker.
(Love you Matt, nobody makes math stuff educational and hilariously like you do.)
@@standupmaths Stop lying! We all know you're future Matt. You're not fooling anyone. Stop bullying past Matt!
@@zerid0 Well he’s definitely lying, he’s the even less distant Matt who can occasionally provide even more corrections.
I love the interruptions its so funny
"log base I don't care" was often the answer I gave in exams
Log, base-eleventeen.
It's imaginary...
I thought the "here's log base #" bit was a bit ha ha for people who already know what's happening but I have a feeling that people who don't already know a lot about logs would probably be scratching their heads. It needed a bit more explanation.
Logging camp is a log base.
Ah, computer science major I take it.
I took 3 years of calculus way back when I was young I don't remember ever covering "log" or "e".
>Matt: this is big O notation
>also Matt: *uses a small o to represent it *
And calls it ‘big zero’ at 15:31
There actually is little-o notation which is like a stronger version of the big-O notation. (en.wikipedia.org/wiki/Big_O_notation#Little-o_notation )
I think the equation on screen is correct, so he should've called it "little-o".
When Matt accidentally implies that his function cannot be growing faster in any way than the function he is talking about at that point even after multiply the function by a constant
Yes, that should be little o. Totally my fault. On several levels.
thats matt's schtick now
"There's a gap between two primes the size of Graham's number. We can prove this exists, first take the factorial."
I spot a problem.
I can help! The factorial ends with more than 7.6 trillion 0's.
Btw Graham's number ends with 7.
There aren't enough theoretical multiverses, each containing our universe's quantity of atoms, in order to write each digit of what you just said on the surface of each atom.
Its odd! But what about using twice the size.
@@anawesomepet how do you know it ends in 7.6 trillion 0's?
@@ERROR-ei5yv for n! there are (Summation from k=1 to infinity of the integer part of n/5^k) trailling zeros, thats how
I absolutely love that Matts WiFi is called “one small step for LAN”
The best on i have seen was
Too fly for a wifi
I wonder if the password is "one giant leap for LANkind". too easy to hack, maybe.
@@vincentpelletier57 It's gonna be a parker password. It'll be as you say except it's arbitrarily misspelled
@@Kram1032 Makes sense
One of my favorites "Rebellious Amish".
I just love the prime gaps sliding over the screen as the video progresses. It's such a nice detail.
"34 OMG!!"
Thanks! I was really proud of that. Fun fact: it was generated in a spreadsheet!
@@standupmaths I'd be surprised and disappointed if it was done any other way.
@@ChrisHarringtonMinneapolis Yes, of the sequence of "largest prime gap up to N," that one is my current favorite.
3 consecutive decades that are empty of primes: 1330's, 1340's, 1350's.
I call it "The Grand Canyon." The "South Rim" is 1327, and the "North Rim" is 1361.
The 8 numbers in that span that aren't divisible by 2, 3, or 5, factor as follows:
1331 = 11³
1333 = 31·43
1337 = 7·191
1339 = 13·103
1343 = 17·79
1349 = 19·71
1351 = 7·193
1357 = 23·59
[Incidentally, today's (2021 Apr 6) Julian Day Number, 2,459,311, is prime.]
Fred
I was unreasonably happy at 9:23 when it became longer than the width of the screen
I just love the idea that Matt spends his free time reading "giant chalkboards covered in math"
No comment.
he's trapped in the Chalk Dimension, trying to calculate a route out.
I bet he also uses his vpn for tracking down dark-web sources of Hagoromo chalk.
@@gcewing You need to go through some really sketchy back-alleys for the _really_ good stuff.
@@standupmaths In fact, this is a comment.
I'll let future Mark finish this comment...
Edit: Future Mark here. Past Mark put me in a bit of a spot since i've nothing to add. Thanks past Mark!
Present Mutt here. Nothing to add from this time period either
Hang on, that's not Future Mark; you're in the past now!
@@vigilantcosmicpenguin8721 were you talking to me? Because to past you i am from the future, so not a lie! 🤣
Time is a social construct
I'm so happy someone still remembers Future Mark. Those benchmarks ROCKED!
Wait, wrong FutureMark... (just search for it here on YT, there are videos of all of them!)
I read the title as (gasp!) and was wondering what was so exiting
same
Primes. Primes are so exciting.
@@standupmaths Exactly! What is more exciting than primes? Nothing. Not even getting a new guitar.
@@standupmaths Hmm.. I think wheel sieve(of primordials) is more intuitive for showing prime gaps. Each successive primordial wheel sieve is made up of its predecessor?
@@standupmaths have you tried taking an unnatural log (log to the base π) of something?
I like that there are GAPS in the video with future Matt interrupting!
The probability that Future Matt interrupts Past Matt is log log n
@@diamondsmasher But how about the odds that Future-Future-Matt interrupts Future-Matt interrupting Past-Matt?
Now we need to calculate the time gaps between these interruptions. Are they behaving primorial?
They are known as Parker gaps
Stealth pun!
Nice to see you and past Matt finally doing a colab, long overdue
So usually there's an enormous wait between new papers released about prime gaps, but suddenly there were two papers released right next to each other?
... Let's call it: "the twin paper conjecture."
Nice.
Hillarious
Do the gaps between papers get larger?
On other fora I've heard "steam engine time" used to mean the moment when conditions are ripe for some innovation to occur, so suddenly a whole bunch of people make the leap at once.
@@vigilantcosmicpenguin8721 its because each paper gets thicker.
"Zeroth things first..." That is the best thing this guy does. 0-indexing is important.
Shouldn’t it be “Zeroth things zeroth”
"As big as it need to be gosh darn it"
Mathematics is a really objective and precise in nature, yes.
It’s precisely as vague as it needs to be
@@poop9267 perhaps "exactly as vague as it can get away with"?
Astronomers see nothing unusual with that statement.
@@SgtKOnyx I think that might be engineering, actually
Hello, I would like to say sorry for my unnecessarily rude comment a long time ago, where I explained how to make the statement precise. There was no reason for me to be condescending while doing so. Forgive me if you remember.
I was really expecting a Matt Parker complicated script writing and timing special where when we were talking about looking for a gap of 8 he would at some point look down and just point at one scrolling across the bottom of the screen "Oh! there's one!"
Funnily enough, there's a gap of 8 at 7:40 (just before he starts talking about the factorial proof) and at 9:30 (just as he finishes talking about it), but none in between
@@LARAUJO_0 is that a gap in the gaps !?
10:54 "840! I mean it's not 87 but it's a lot smaller." Lovely Parker sentence.
I came looking for this, sortof. @ 9:49 he says the gap is 89-97. Then mentions 87 at ur stamp. I was confused, and now im More confused cuz apparently i missed a joke too... :(
840! is way bigger than 87
Wasn't 840!, but rather plain 840, which is much smaller than 8!
To be fair, you need to have a really high IQ to predict the date of the next Rick and Morty season
I must have a really high IQ then, because I know the date of the season 5 premier.
@@blindleader42 it is easy for small numbers (1 to 5) cause you can brute force it with google. Mathematicians are still unsure on values as small as 6 though.
@@yyeeeyyyey8802 OK. I predict season 6 sometime in 2022... or never.
Would you believe they announced the date between me filming this and release it. You’re welcome.
I can find a lower bound on the date. But it's not very impressive.
People will think I'm strange now when I'm working my exams and I whisper "Future Matt? Any help on this one?"
Oh man, Yeah! Future Matt - hear our prayers! Answer our math/s questions and elevate the quality of our calculations!
Now my suggested videos include: “Making a log carving robot”
Following that channel keeps me happy
should have been used private internet access (tm)
Lol, someone's (ro)bot isn't intelligent
By the way, changing the base of a log only scales it by a constant amount. That is, log_a (x) = c * log_b (x) where c = 1 / log_b (a).
So for _any_ log plot, changing the base of the log would not affect the shape of the plot. It just changes the scale of the plot.
Use base 1 or 0
@@happygimp0 oof, infinite and zero scale
👏👏
Even easier to see it using the change of base rule
log_b(a) = log_x(a) / log_x(b).
The divider is constant for all different values of a
Nice ME system you got there.
@@cubixthree3495 Thanks :)
Drinking game: take a shot everytime a gap of 2 appears at the bottom
younger matt starting at 487 saved lives
If you and your mates (who are betting on another numbers) are cursed with immortality, you'll be the most sober guy in the room.
>Big Zero
chungus*
@@philkaw say chungus but replace the "chu" with "amo"
Amumugos
Amonges
Amongus
"In this case is 840. I mean, it is nt 87, but it is a lot smaller"
- Matt Parker
I love out of context quotes.
Matt at 0:33: "Because they're all odd numbers…"
The number 2: 🥺
yeah, definitely an odd prime for sure.
and the gap between 2 and 3! they're consecutive primes too!
yet there's no point eternally in the bottom left corner of all of his graphs for the single gap of 1 that appears
@@DagothXil Speaking of gaps, is it actually relevant to say there's a gap between consecutive numbers?
"Big zero" spotted! Glad you, the author of Humble Pi, left it in.
Looking at the graph, I have my own conjecture about the primorials/jumping-champions connection but I don't know if it's been considered already.
As Matt points out at 19:45, the top of the line is all the multiples of 6. The ones he highlights as suspicious contenders, who are raised slightly above the others, are all multiples of 30 until 210 which is raised even more from the other lines.
My suspicion here is that the 'thickness' of this line is actually the result of multiple lines being overlaid, with each line sharing the same common factors.
So one line for powers of two, one for multiples of only 2 and 3, for 2,3 and 5 and so on. In the Silva paper in the description, they highlight the multiples of 6 in another colour and I think it would be interesting to see the same for the rest of the primorials which, by their definition, would be the lowest value for each of their respective lines.
Each line then, is more popular than the last as numbers grow higher but lower numbers are more frequent for any given line, which is why it takes time for each champion to jump to the top.
As an afterthought, this might explain the bumpiness of the lines, too. There are sets of unique prime factors that are non-primorial (ignoring the odds) - 2*5, 2*7, 2*3*7 and so on. From that we would expect bumps at 10, 14, 20, 28, 40, 42... At least up to that far, the graph looks to me like it meets expectations.
I've dipped the tiniest tip of a toe into the deep lake that is prime number theory, and what most gets me is just how simple and breezy this video can come off as, all the concepts being so easy to explain, yet underlying them is no doubt some extraordinarily complex mathematics.
Very true, best example is the paper containing the proof of the ternary golbach conjecture lmao.
It's around 11:18 where i stopped watching a math video but started watching a magician's performance.
"... because they are all odd numbers the gaps are always even..." 1 not being a prime i could accept but now 2 is also left on the side that i can not allow!
I don’t know it’s the only even prime, hardly fits in with the others ;P
@@Lanthardol as math teachers like to jest, 2 is the oddest prime of all.
It's interesting you should point this out: because the only reason 2 was declassified as a prime was convention - to avoid having to say "any prime except 2" or "take any odd prime". In this case it avoids having to add the qualification "all prime gaps, except the gap between 2 and 3, are even.
@@ig2d there isn’t a gap between 2 and 3
3 minus 1 is 2. If we take 2 out of the primes club, can we bring 1 back in?
16m42s: "A day later, on the 21st of August, 2014, someone else proved the same thing a different way."
[Shows title & Abstract of a paper by James Maynard.]
Hey, he's not just "someone else;" he's that famous prime-o-phile from the Numberphile channel!
Fred
Holy crap the editing these videos must take. Aside from the enthusiasm, I have a lot of respect for the time and effort you put in. Thanks Matt!
Matt: "Anything I say from now on assume it's a sensible case"
Us: No, I don't think I will
Ooh, time for my favourite maths joke!
"What sound does a drowning number theorist make?"
logloglogloglog...
i almost ordered a custom t-shirt with that printed on it, its my favourite joke too
“It’s called Big G, because it looks for big gaps” 😂
I imagine Big G is a gangster boss
@@3Ppaatt my thought exactly 😂
"I'd like you to meet Big G from Chicago."
You have no right being this funny and simultaneously educational. I love it.
If this youtube thing doesn't work out at least we know you have the pointing skills to be a weatherman
It's not big O notation, obviously its a small o. Small o is much stricter than big O.
If f in O(g) it means that f(n) will be smaller than a constant times g(n) after some n great enough
If f in o(g) it means that f(n)/g(n) tends to zero as n tends to infinity.
So while both are Landau notation, big O acts as a ≤ while little o acts as
Thank you.
Was gonna comment this ty
There is actually a seminar by Terence Tao on prime gaps uploaded to UA-cam by UCLA from just after they published their papers. It provides some cool insight into what happened at the time.
I somehow missed that. Will check it out. Tao is amazing.
Matt: So I've written some Python code...
Matt's Laptop: pleeez haalp
20:18
I was very proud of myself when I'd predicted "Oooh, the next peak will be at 2310 because that's 210*11, and 210 is 7*30 !" a few seconds before he mentioned this.
210 != 7*30! 😉
@@RedGorillaa Yes it is. Use a calculator.
@@smergthedargon8974 You've been foiled by the unintentional factorial.
@@smergthedargon8974 7*30! = 7*30*29*28*...3*2*1 != 210 😉
@@Euler13 Oh, so you're just being a smartass.
7:31 "Arbor Terry" Love that guy. Always planting trees.
Thank you so much for explaining the functions! I've seen other functions before but couldn't understand their meanings. You made it so much easier! Great job!
I love these videos! They always make me confused since i didnt have an oppitunity to study maths past my GCSEs, but it all so facinating from what i can get
One small comment: The papers seem to use little O notation, not big O. The difference is that the bound is strict.
Suggestion for your 1M subscriber special: complain about all the times past Matt wasn't excited enough about graphs or maths in general. That was fun!
Honestly just some of the best STEAM communication I'm subscribed to; I just love the enthusiasm and passion and humor.
Thanks Matt for finally making a video on this topic! I have been waiting patiently for this video :-). Absolutely love your channel!
The Rick and Morty comparison is something I didn't know I needed today.
Not all prime numbers are even. 2 became prime against all odds.
ALL. BETS. ARE. OFF.!!!
Beautiful animated scatter plot of how the prime-gap changes. Thanks for making my day.
fantastic video! I applaud the video editing. When you pinpointed the individual points on the graph with your finger (the ones that take the lead eventually for common gap size), I have no idea how you were able to do that . And the running timeline at the bottom was great, something extra to look at
As the "top point on the line" increases from 6 to 30 to 210, etc the shape of the line doesn't change. The resolution of the plot gets very much smaller and the earlier, smaller, numbers are just smushed into the band under the top point. As 2 is when the top point is 6.
I was going to say, since the bottom right is roughly (ln (no. of primes))^2, it will continue on WAY faster than each next primorial taking over. However, when all 150 million were animated starting from small numbers, the slope of the line definitely looks like it drops with more and more primes.
Also, I think Matt should try skipping a large amount of the first primes to make these calculations, such as going from the 140 millionth to 170 millionth primes.
Future Matt appearing and scribbling everywhere gave me Emperor’s New Groove vibes
My gosh, the primorials fact blew my mind, it's crazy how clearly there must be some underlying structure to the primes, and how much it brings about such neat patterns, yet it completley illudes us.
Excellent video!! I never thought about the shape of that 'line' with such a huge numbers taken into consideration, but that is actually a great question!
its worth noting that when the base of the logs change, the scale of the plot changes as well. its not the same number, but its just scales the axis
He also pulled a sneaky Y-axis flip for 0.001, it started rising in the negative direction
@@ilurv2eetpie yup, though you could argue that this is just scaling as well
The video I watched before this was a video about Rick and Morty and I'm not sure if the algorithm is just that good or if an amazing coincidence just happened
Blame future Matt.
YT does not make coincidences, I mean what a mistake... Shoot, this is going nowhere...
I'm so disappointed at how good that Rick and Morty joke is, because I don't have any maths friends that would appreciate how spot on it truly is.
@sixequalszero except you already heard the joke
You must be the one to convert them to mathematics.
As Matt exemplifies in his presentation, time for pure mathematicians is merely the succession of numbers. He constantly refers to the gaps getting bigger "quickly" as the number X in the lower boundary equation gets bigger. What an educator! I've been enthralled from beginning to end. Thank you!
This line is hipnotizing me
hi matt! apologies for this probably long comment! firstly, i absolutely love all of your videos you have such a way of telling mathematical stories without losing any of the maths itself which i love so much! i, and i think some other people online, have noticed that you often will use singular they/them pronouns for people and according to reddit this is also true for much of Humble Pi. I thought this was cool! after also hearing a professor of mine (physics, so i was asking about how the uni may try to better express that people who use gender neutral pronouns are welcome in this area) discuss the use of gender neutral pronouns by academics (something i still haven’t fully been able to understand, maybe just for ease? or confidentiality?) this is what i had just assumed was what you were doing. And then this video! at 14:11 you referred to past matt (which in some way is you but i don’t do philosophy) with they/them pronouns! which i, again, thought was very cool. i can’t find anything online about you discussing your gender and obviously if this is something you’d rather not explicitly discuss because that is your personal life then that is very cool and understandable. i don’t really? have a question um i apologise if this has been a waffle i just wanted to see if you had anything to add onto this, i am nonbinary and really appreciate this sort of stuff of moving to normalise the use of gender neutral pronouns. especially in stem fields!! ❤️
To be honest: I respond to any of he/him/they/them and don’t mind anything else as long as it’s not malicious. I actually refer to myself as sometimes they/them for the same reason I do other people much of the time (and 100% of the time if they are hypothetical people like the examples in my book) which is to normalise non-gender-specific language. I hope that makes sense!
@@standupmaths absolutely! thank you for clarifying and responding! i think what you’re doing as a maths educator and curiosity-inspirerer(?) is so wonderful
@@standupmathsVery cool.
Superb Video!
I'd a similar thought brewing in my head not long ago. That said, this both helps fantastically, and also illustrates it better than I might have, given other things to focus on.
I kinda like the edits, to clarify. It has a nice pace to it, and you addressing your past self is quite funny.
I couldn't avoid getting distracted every time twin primes appeared
For clarification, whenever someone refers to log without a base, it is ALMOST ALWAYS log base e (or ln).
This is so great! One of my favorite UA-cam channels ❤❤❤❤
6:20 The animation has the horizontal axis labeled with half the gap, but you can tell by the multiples of 30 and where they stay higher on the line that it's actually scaled by the gap instead of half the gap. At the very end of the animation, yes, the scale suddenly changes to half the gap.
nitpicks! at 12:46, you use a little o for big O notation - thats kinda confusing because there is a little o notation, which one are you talking about?
all the linked papers in the description use little o, so i'd assume he actually means little o.
@@randomdude9996 Yeah, I think it should be little o. Otherwise there would be no point having 1 in "1 + o(1)", as 1 + O(1) is just the same as O(1).
20:46...we managed to “prove” that it “implies”... 😆 I love these.
Can I just say that I appreciate the "bonus" of the continuous prime-line that keeps going on the bottom the whole time? :)
Mate... That animation at 6:30 is brilliant.... Seriously well played!
You might want to check out the video I just made (look for the one and only video on my channel).
It's very bland - no sound, and only comprehensible if you saw Matt's video. But it's an extended version of that animation at 6:30 :-)
0:34. "Because they're all odd numbers". The Parker Two
3:55 No, it‘s not! The probability that a number is prime is 100% if it‘s not a multiple of any number below it except 1. If it is, then the probability is 0%.
I’ve seen a lot of people get worked up about this. Interesting!
In the part where you do 8 factorial and then reduce it to the greatest common multiple, you could just use primorials. The reason this still works is that the product +2, +4, or +8 all are composite because 2 divides into them. So you actually wouldn't have to multiply 2 three times, but just once.
I kinda love that these big numbers you're talking about (like, 10^|my overdraft|) are infinitesimal fractions of huge numbers like Graham's Number and Tree (3), which are themselves, by definition, infinitesimal fractions of the entire number line. It blows my mind that mathematicians can construct and manipulate such big numbers, while simultaneously recognising that these numbers are trivially small.
For the first three or four minutes, I was wondering if you were heading towards the Riemann Hypothesis, but then you went somewhere I wasn't expecting.
Since the log base doesn't matter, the graph should be animated such that the log base is always the frequency of gaps of size 2. That way the animation will always grow from 0, and you have an absolute reference point.
What rolls down stairs
Alone or in pairs,
And over your neighbor's dog?
What's great for a snack,
And fits on your back?
It's log, log, log
It's log, it's log,
It's big, it's heavy, it's wood.
It's log, it's log, it's better than bad, it's good. "
Everyone wants a log
You're gonna love it, log
Come on and get your log
Everyone needs a log
Log log log
To what tune do I sing this?
(this sent me down a rabbit hole of the evolution of the Slinky ad--the jingle originated in the 60s, but "without a care" became the better-rhyming "alone or in pairs" in the 70s)
That "Ooh matrices" at 24:10 was so in-character
Just a reminder:
For expressions like log log log ... log x, one can always use the recomposition notation: $\log \overset n \circ x$, where n is the number of logs.
Another reminder: awesome video!
I was hoping future Matt would keep interrupting after the second one. I was not disappointed.
Me whenever I remember that 2 is prime:
*You may be on this council, but we do not grant you the rank of master*
Who does number 2 work for?
I love the editing
Alright, that opening was amazing.
You said big-Oh notation at 12:44, but just to clear that is a little-Oh (which is also a type of big-Oh notation), right?
Oops, I just left a comment asking exactly the same thing before seeing this
He later calls it Big-Zero, but then says it gets smaller as x gets bigger, so I think it is supposed to be a Little-Oh
Little O means that your function can’t grow faster than any function even after multiplying the function you are comparing.
In practice, it means that to be little-o of a function means you really grow slower than a family of functions (as opposed to big-o meaning to grow slower than or at the same rate as a family of functions).
The way I think about primes from a non-mathematical perspective is that they are recursively self-destructive.
What I mean by that is that for every prime we find, every future multiple of that prime can no longer be a prime (self destructive and therefore recursively defined), and every prime is therefore defined by the *lack* of a prime divisor of itself earlier in the sequence. The more primes we find, the larger the future gaps will be because every new prime removes infinitely many future potential primes.
These gaps *would* increase linearly with the primes if it weren’t for the fact that the multiples of primes *overlap* with each other at an increasing rate as more primes are found (not sure how to word this better), thus giving us a *logarithmic* increase in the gaps instead of a linear one.
Another way to look at the logarithmic nature of prime gaps is from the fact that when verifying that a number is prime, you never need to check any prime higher than the square root of the number you’re checking. In other words, as the number you’re checking increases linearly, the amount of numbers that could prevent it from being a prime only increase logarithmically, and thus the number of gaps can only increase logarithmically.
This recursively self-destructive definition of primes is part of why primes are so difficult to get a solid grasp on. Every prime’s very existence is defined by *not* being a multiple of a previous prime.
This is a very good explanation. Thanks!
THANKS! Thanks for someone finally mentioning Primorial Numbers. They are so interesting, especially when looking at Primes. Thank You Thank You Thank You
I like logs too! A log house is long-lasting and cool, you can make wood statues out of logs, logs have an industry of their own! Logs are just so amazing and useful.
Is there a function f(p) = k, where p is prime, and the next prime is ≤ p+k ?
(i.e. an upper bound on the next gap, in terms of the size of p)?
You've sent me down a fun rabbit hole reading the prime gap wikipedia page. Anyhow, Bertrand's Postulate states that there is always a prime number between n & 2n for n > 3. So f(p) = p works.
@dejadee has a good answer there. The proof of Bertrand's Postulate doesn't look particularly straight forward: en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate
The first thing that came to my mind is using the classic Euclid proof of infinite primes to get a very inefficient upper bound.
In short if p_n is the nth prime, N = p_1*p_2*p_3*...*p_n +1 is either a prime or divisible by a prime larger than p_n.
So the difference between N and p_n is an upper bound on the gap.
f(p_n) =p_1*p_2*...*p_n + 1 - p_n
0:08 That's why he's so smart!!
"As a number theorist i have a favorite numerical sequence. Did you know that if you take the number 41 and add first two, then four then six etcetera. To get the sequence 41,43,47,53 etc. That the first forty numbers are all primes. And that no similar numerical sequence of that lenght exists." - General Michael O'Toole. RAMA the video game. Based on the works of Arthur C Clarke and Gentry Lee. :)
15:10 „the whooole thing here is bigger than a regular log“
thanks Matt
I was just gonna search this up...
The video is oddly quiet
I wish i could watch thus channel while learning in a middle school. I envy nowadays students have this opportunity.
I've never been so disappointed in Past Matt. Many thanks to Future Matt for being so awesome.
The biggest prime gap you will see scrolling by the bottom of the screen is 34. To see it, just go to 16:35 :)
I wonder if the time stamps of the interruptions of "editing Matt" are somehow related to a sequence of prime gaps?
Well, now I wish they were.
We need a script to plot the gaps.
I love the prime line just moving along with throughout the whole video.
All this talk about prime gaps reminds me about runs of sequential Collatz sequences with the exact same length. Blows my mind!