Bonus fact: 10^10^10^34 was found ASSUMING the Riemann Hypothesis was true. In 1955, Skewes found another number (10^10^10^964) that was without the use of the hypothesis.
I've never understood why people use log(x) with base e. ln(x) is shorter to write, and people won't mistake or get confused on whether log is with base e or base 10.
@@poisonoushallucinations3168 well, we are supposed to only use ln(x) for e, log(x) for ten, and log-subsript n-(x) for base n, so we have a solution, but apparently not everyone follows the rule, leading to confusion. Like grammar, but maths!
alan smithee log(x) isn’t for 10 though. It’s been arbitrary for quite a while, with 10 and e being the more common bases. The newer notation lg(x) for base 10’s there to help to avoid confusion when using log(x) without specifying a base
They don't mention it in the video, but Skewes' result assumes the generalised Riemann hypothesis. Without that assumption, Skewe's upper bound is 10^10^10^963. Quite a big bigger.
+Giggstow No, if the first instance of the inequality flipping was over 10^10^10^34, that would in fact disprove the generalised Riemann hypothesis. As the last part of the video shows, we already know that the inequality flips at a much smaller number than 10^10^10^34, but that's not proof of the Riemann hypothesis either, because it's inductive, and proof needs to be deductive.
That number is the actual number Skewes found but writing it (or a close enough approximation) with 10’s as the bases is slightly easier to comprehend.
So, if they'd had Numberphile in the 70s, they could have had Skewes himself talking about Skewes' number, just as they've had Graham talking about Graham's number. Don't get me wrong; Asimov would have been a great guest, but he wouldn't have been able to talk about it the same way as the mathematician himself.
Compared to TREE (3), Graham's Number is practically 1 Compared to loaders number, TREE (3) is practically 1 Compared to typical busy beaver numbers, all of the above are practically 1
@@blue9139 There is a function called M(n) I saw in numberphile' TREE(3) video. It i not Mersenne prime function, but it works like this: M(1) = Largest number one mathematician can define by working for a year in perfect harmony M(2) = Largest number two mathematician can define by working for a year in perfect harmony M(3) = Largest number tree mathematician can define by working for a year in perfect harmony ... Yes. It is extremely unexpectable and extremely big, bigger than ANY NUMBER ever defined. In the process of calculating the result, countless interesting notations will be made.
I had no idea that the concept behind Skewe's number was so simple. So I just ignored it., thinking it would not be worth the effort to understand it. And you knocked it flat in ten minutes. Well done!
I love how James can really make me grasp something that's way over my head-enough to understand-without making me feel like an idiot-even when he needs to use math that's far nehind my capability.
And here you have somebody who was fortunate enough to have been tutored in mathematics over 30 years ago (and paid for it by reading and describing to him the Modesty Blaise cartoon every day) watching your beautifully done explanation and remembering Stanley's enthusiasm for mathematics. Well done.
On reflection, a few years later, what I love about this number is that it's the upper bound of an approximation of an indirect observation of an inequality of an approximation for counting primes. Such beautiful indirectness
Sometimes,I watch number phile even though I don't get anything because seeing the guy getting excited and enthusiastic about explaining, is somewhat fun.
I know I just commented about this about 9 minutes ago, but it needs to be said again: *_JAMES GRIMES' ENTHUSIASM FOR MATHEMATICS IS JUST ABOUT THE PUREST THING ONLINE._* P.S. (takes a long drag on a cigarette) Smoking is baaaaaaad.
It made me so excited that you talked about Isaac Asimov. He's my favourite person. You know that question "If you could spend an hour with one historical person who is no longer alive, who would it be and why?" I used to not know how to answer that question. But these days, I know it would be Isaac Asimov, without doubt. Knowing that he wrote about this number has made me realize that I need to read more of his non-fiction stuff, like his science articles. I started reading a book of his articles once, at my school library, and I loved it! But for the most part, I've only been reading his short stories and novels. Time to scour the internet for his science article anthologies!
I actually understood the formula he explained perfectly fine. The one where he said that "some of you won't be familiar with what I'm going to do." I feel like my studies have gone pretty well.
This is the Numberphile vid I've rewatched the most times. I always return to see the smoking '70s Grime, but then I remember what a cool concept is actually being discussed.
3:50 Log(x) is normally x's logarithm--in base 10 in this case since no specific base was given (as far as I've been taught); ln(x) represents x's natural log and is in base e, of course... That or some conventions have really evolved in the last 20 years...
8:48 : The use of an "x" as the multiplication symbol bothers me. It's even more unforgivable as it's written with a serif font and doesn't even look like a simple cross anymore.
This is so amazing. I watch again and again. James is so wonderful as a teacher. He also says how it would be if Isaac Asimov talks about it on 70s Numeberphile. :-D He is superb this guy.
+Flandre Scarlet I may be wrong but since both functions are continuous for large numbers, the intermediate value theorem should be usable to show that there should be one specific number (exactly where the equality sign flips) that gives equal values for both functions. I imagine it would be a number with a long if not infinite tail of decimal places. In short, as x increases, the functions have to get closer and closer together before the equality sign flips, very briefly becoming equal as the flip happens.
Thanks for the info :) Just to be clear I was referring to the approximation pi(x) = x/ln(x) which is continuous for x < 1. If there are points for which pi(x) < li(x) and for which pi(x) > li(x) then there has to be a real number in between where pi(x) = li(x) for a specific real x. That is what I assumed was being asked.
In case anyone else was confused round 8:00, no, I'm fairly sure the co-writer of How I Met Your Mother didn't co-write "A new bound for the smallest x with pi(x) < li(x)".
Wait, wait, wait now. You skipped something important. You claim that we know a run of integers where this inequality is flipped, but we don't know the first time it flips. So that means we somehow know how many prime numbers there are under some numbers, but not under smaller numbers. This requires an explanation :p. Please explain.
I think the issue is that the way we calculate pi(x) for large x is by using li(x) and the known error formula which involves related functions of li(x). And calculating li(x) for large x is not an easy task. This approach was formulated by Reimann and is greatly connected to his zeta function, so much of the discussion of the solution to this problem revolves around assumption of the Reimann hypothesis.
Big fan of Isaac Asimov; I definitely recall reading the essay JG is referring to. IA described it as the largest number usefully applied to a proof at that time.
+Cristi Neagu Yeah the notations of logarithms are weird. ln(x) is usually the natural logarithm to base e, never seen something else. But then there is lg(x) which is either base 2 or 10, lb(x) and ld(x) usually base 2 and log(x) really depends on the context. Its often base 10, but it can be anything.
+Cristi Neagu In advanced mathematics, natural logarithm is conventionally detonated as "log(x)", since there is absolutely no need to use logarithm in base 10. If you ever need to use log base 10 (which you will probably very rarely do in advanced mathematics), you may just write: (logx)/(log10) (this is due to base change formula). Conversely, In biology, astronomy, or engineering, natural log is almost absent and therefore log(x) will indeed refer to base 10 log.
+KevinJRattman For exactly the same reason, we almost never wrote "log" in my engineering studies (since there's no need for logarithms in base 10), but used "ln" for almost everything. "log" would mostly be used for the extended, complex-valued, version of the natural logarithm.
+Cristi Neagu In some contexts, ln(x) is denoted as log(x) where it's clear that the natural log is being used. For contexts where it's less clear, the notation ln(x) is used instead.
The better way of saying what Greg said is that those functions aren't continuous, they're discrete, since you can only plug whole numbers into them. So no, there isn't necessarily a point where they cross.
+Brendan Beaver Functions that are defined on discrete spaces are necessarily continuous. It's a basic topological concept, since every subset of a discrete topological space is open.
+hpekristiansen because they might truly have very close values, but we cannot say the values would be exactly the same at a certain point merely because the signs flip, since we do not know if the functions are continuous. Check out the intermediate theorem.
I always wonder, does he have all the numbers written down somewhere on paper outside the camera or does he just know these numbers by heart? He seems like such nice guy! Great video as always!
9:19 and it APPEARS that it’s gonna HOLD FOREVER AND THEN… it flips 😂😂
9 років тому+4
Have they changed nomenclature? I learnt that, if we are writing numbers on base 10, log means log on base 10, not on base e as they use on the video. Log on base e is normally written as ln.
+ben1996123 In pure mathematics that may be true, but in science and engineering ln() is commonly used to denote natural log, where otherwise one would have to write either log (subscript) e or log (subscript) 10 to avoid ambiguity. (In science and engineering, clear, accurate communication is essential, and every item of nomenclature must be defined where used. In this arena some forms of shorthand notation have become universally adopted for the convenience of all concerned.)
+Víktor Bautista i Roca log is the traditional way to write natural log. Base 10 logs are easier to teach, so you learn those first, and we just decided to switch the usage of log to log10 for education. Now it's kind of muddled and recommended that you specify somewhere which is which.
There's a little bit of fudging going on here. James only uses the integer result from the Li(x) function and says that Pi(x) appears to always be less than. Under those conditions the rule actually fails for Pi(13): Pi(x) = Int(Li(x)) (=6) However, while James says "always less than" he actually contradicts himself by writing "less than or equal". Is he trying to have his pie and eat it, I wonder? Either way the pie looks a little bit sloppy ;-)
How could Stanley Skewes tell that the inequality flipped? I mean the Pi-Prime function has no closed form to this day, hence the notion for approximation right? So for recognizing the flip you would have to calculate and count all the primes of 10^10^10^34 by foot (or computer). But that doesn't sound like a task that could be done by computers yet, or at least by the computers of 50-100 years ago when he did this work. Do we have any genius here to resolve this question? ;)
ben1996123 Not sure if trolling or just not eager to be helpful. I would really like to know. As a computer scientist, analysis and co are not my prime fields of mathematics.
Patrick Fame neither. i told you. analytic number theory. but don't expect to understand it unless you do a phd or something. the largest value of pi(x) known is only pi(10^26) which i think took about 15 cpu years and 128gb of ram to compute.
+Laatikkomafia In english big numbers aren't called the same as they are in your language (and mine as well). So he's not wrong, but I get how it can screw up your head.
Gauss did it it when he was 15 ! and I am already 16 and still at high school. I am hats off amazed and saddened at the same time. How can someone be so great! Is it there surrounding or their unique enthu or their natural intellect or is it something else?
My number: The biggest number that will ever be found + 1. If you find a greater number than my number, it is still gonna be one bigger than yours. Checkmate.
Skewes Number is very, very small compared to Graham's number nowadays, like it was back in the 19th century when people thought no one could comprehend this number. But in reality there are many big numbers now so Dr James explaining this must have been surreal.
Made me think of massive stars collapsing on themselves, shrinking in volume, passing that "flip" boundary in terms of density and then basically breaking fundamental laws of the universe.
if this video has X views, 42^69^420^X ~ 10^(1.6*69^420^X) ~ 10^69^420^X (multiplying something larger than 10^10^100 by 1.6...) ~ 10^10^(1.85*420^X) ~ 10^10^(1.85*10^(2.6X)) ~ 10^10^10^(2.6X+0.25) ~ 10^10^10^(2.6X) ~ Skewes' number when this video has 13 views, and 10^10^(2.21*10^1985999) as of when this comment was posted. when this video had zero views you get 42^69^420^0 = 42^69^1 = 42^69 ~ 1.01*10^112
At 6:50, James says that the inequality sign flips infinitely many times. Does that mean that there is a direct relation between Li(x) and pi (x)? Meaning it's not just by chance that Li (x) delivers a value close to pi (x)? If the relation was accidental, wouldn't it verge off at some point? Can we learn something about prime numbers from Li (x) if we understand this relation?
I've come up with a method for estimating the number of primes less than a number that gives perfectly accurate results (though it is a bit computationally intense). For the number of primes less than the integer n, starting with 2 compute the prime factors of the number and there are any factors besides the number itself and 1 then decrement your count (which starts at n) by 1. When you reach your number n your count should equal the number of primes less than it.
Skewes' number is basically 10 to the power of 1 with 10 decillion zeroes after it. To better explain how massive this is, googolplex is 10 to the power of 1 with 100 zeroes after it. That means googolplex is 10 to the power of 1 with 100 nonillion zeroes after it times smaller than Skewes' number. To calculate that difference, the quotient of Skewes' number to googolplex is equal to 10 to the power of googol to the power of 1 nonillion.
I started out being mildly curious about Skewes' Number, but this quickly turned to intensely wanting to see a 1970s Numberphile episode starring a chain-smoking James Grimes with a super-wide tie and epic sideburns
I love his enthusiasm so much!
+pyromen321 me too!
+Slaughter round i first thought pyromen is talking to himself :D
+pyromen321
Have you ever noticed how large his pupils are in every video? I think his enthusiasm is somewhat medicated.
+Marc Tißler For half a second I thought pyromen was talking to "been there". *facepalm*
+ty_ger Nah he just has dark eyes. Beautiful, dark eyes.
"1899 - 1988"
This is great.
+Horst Kevin von Goethe Makes me feel a little OCD, but that was the first thing I though of.
+Jordan Lagarde yea... he could live from 1888 to 1999... that would be insane :D
+Peter Bočan Nah. From 1111 to 2222.
+LightningCat Craft Wouldn't he be dying a little young?
+Akașșș You are correct 1111 years is a very little time to life :(
Bonus fact: 10^10^10^34 was found ASSUMING the Riemann Hypothesis was true. In 1955, Skewes found another number (10^10^10^964) that was without the use of the hypothesis.
I've never understood why people use log(x) with base e. ln(x) is shorter to write, and people won't mistake or get confused on whether log is with base e or base 10.
I didn't know people did that, and now that I do, I am sad. :(
Solution: Don’t use log(x). Just ln(x) for base e and lg(x) for base 10
@@poisonoushallucinations3168 well, we are supposed to only use ln(x) for e, log(x) for ten, and log-subsript n-(x) for base n, so we have a solution, but apparently not everyone follows the rule, leading to confusion. Like grammar, but maths!
alan smithee log(x) isn’t for 10 though. It’s been arbitrary for quite a while, with 10 and e being the more common bases. The newer notation lg(x) for base 10’s there to help to avoid confusion when using log(x) without specifying a base
@@poisonoushallucinations3168 well, you can take it up with the a level boards, I won't argue on their behalf.
I would love an "R rated" numberphile that assumes complete knowledge of Calculus or the like
+Michael Marks I think that's what the Numberphile 2 channel is for.
Michael Marks lol
"Numberphile: Adult Swim"
Michael Marks I would leave a like but I did bad in calc so
@Michael Marks I don’t know if I would be able to show that to my children, they’re too young!
It's been a while since the last James Grime video he's my favourite. He's just so happy and optimistic and an incredible explainer.
Dr James talking about the 70's *"Numberphile"* is just amazing! He is very enthusiastic!
10^10^10^34 has 10^10^34 digits.
10^10^34 has 10^34 digits.
I think "trillions and trillions of digits" is a bit of an understatement.
Ian 07 off by 1, but close if you're rounding. 10^n has n+1 digits.
True
So it's a 1 followed by 3400 zeros (less than a trillion digits long)?
2s7a2m7 the number you just described is how many digits there are in swewes' number
2s7a2m7 dude 10^10^34 has 10^34 zeroes after it.. which alone is more than a trillion. 10^10^10^34 is mind numbingly larger than that..
They don't mention it in the video, but Skewes' result assumes the generalised Riemann hypothesis. Without that assumption, Skewe's upper bound is 10^10^10^963. Quite a big bigger.
+Giggstow No, if the first instance of the inequality flipping was over 10^10^10^34, that would in fact disprove the generalised Riemann hypothesis. As the last part of the video shows, we already know that the inequality flips at a much smaller number than 10^10^10^34, but that's not proof of the Riemann hypothesis either, because it's inductive, and proof needs to be deductive.
+Giggstow no, the implication is only one way
+Luke Shirley Modus tollens would like a word with you.
+pyropulse Nah mathematical induction is a form of deductive reasoning.
Persona GRH is not the contrapositive of that statement.
I'm much more interested in the "e^e^e^79" number shown in the excerpts of the paper
That number is the actual number Skewes found but writing it (or a close enough approximation) with 10’s as the bases is slightly easier to comprehend.
ARG!! you * _ * miuf!
Skewes lived from 1899 to 1988. Thats interesting in itself
No, it isn't.
Tony Bates
Matt Parker would fight you for that :^V
Tony Bates you don't decide that
So, if they'd had Numberphile in the 70s, they could have had Skewes himself talking about Skewes' number, just as they've had Graham talking about Graham's number.
Don't get me wrong; Asimov would have been a great guest, but he wouldn't have been able to talk about it the same way as the mathematician himself.
That's so not interesting. Unless the approximate birth of Christ and our arbitrary calendar that followed from it mean anything to you.
I especially like the videos with Dr. James Grime. His genuine enthusiasm and passion for what he's talking about makes the topic very interesting!
Compared to Grahams number, that number and 1 are virtually the same.
Compared to TREE (3), Graham's Number is practically 1
Compared to loaders number, TREE (3) is practically 1
Compared to typical busy beaver numbers, all of the above are practically 1
Christopher
Yea lol. RELEASE DAT OBVILION
This reminds me of one of Carl Sagan's quotes in Cosmos (Ep09):
"In fact, a googolplex is precisely as far from infinity as is the number 1."
@@christopher9624 For every positive number n, there is a bigger number m for which n is practically 1.
@@blue9139 There is a function called M(n) I saw in numberphile' TREE(3) video. It i not Mersenne prime function, but it works like this:
M(1) = Largest number one mathematician can define by working for a year in perfect harmony
M(2) = Largest number two mathematician can define by working for a year in perfect harmony
M(3) = Largest number tree mathematician can define by working for a year in perfect harmony
...
Yes. It is extremely unexpectable and extremely big, bigger than ANY NUMBER ever defined. In the process of calculating the result, countless interesting notations will be made.
Wow. I can tell JUST by the tone of his voice how excited he was about Graham's number. I am so jealous.
1:50 JUST DO IT!!! DONT LET YOUR DREAMS BE DREAMS!
+ＲＸＱＺ they really should.. but without the smoking.. it's not the actual 1970s anymore and smoking should not get advertised like that ever again
+whoeveriam0iam14222
Maybe after a googol googol googol years it SHOULD get advertised again? Stop extrapolating.
+Simo Vihinen The universe might have stopped and started a few times again since then so yeah maybe it should be advertised then, who knows.
+whoeveriam0iam14222 Come on man, don't you wanna be cool?
***** I asked you first.
After this discovery, Skewes sketched up the first known rules for Flipadelphia.
The Hoax Hotel flip flip flipadelphia.
Love Numberphile and Dr. Grime!
I had no idea that the concept behind Skewe's number was so simple. So I just ignored it., thinking it would not be worth the effort to understand it. And you knocked it flat in ten minutes. Well done!
James, this is awesome! I'd love to see 1970s Numberphile. I love huge numbers like these.
Always a joy to see you, guys, even if there's absolutely no chance to understant a thing what this all about.
I love how James can really make me grasp something that's way over my head-enough to understand-without making me feel like an idiot-even when he needs to use math that's far nehind my capability.
Sooooo frigging good! I wanna see 70's Numberphile!!
"AND IT APPEARS THAT THIS INEQUALITY HOLDS AND THEN .. it flips."
Man Numberphile makes math seem like something sweet to study, too bad nothing like this is taught in college, not even in the fourth year.
Crunkmastaflexx
I feel like numberphile is really dumbed down...
I... learned about Graham's Number, Skewes' and TREE at 5th grade...
I... learned about Graham's Number, Skewes' and TREE at 5th grade...
TREE(3) Please do a video on TREE(3).
Yes please
+J.R. Trevino I've been super curious about this. TREE(3) is apparently bigger than Graham's number.
+Jordan Shank yes it is, apparently Grahams number in unnoticeable by comparison
YES! YES! YES!
Loader's number is nothing special, it's just the output of a computer program.
And here you have somebody who was fortunate enough to have been tutored in mathematics over 30 years ago (and paid for it by reading and describing to him the Modesty Blaise cartoon every day) watching your beautifully done explanation and remembering Stanley's enthusiasm for mathematics. Well done.
I wish I was as happy about anything in life as this man is about math.
On reflection, a few years later, what I love about this number is that it's the upper bound of an approximation of an indirect observation of an inequality of an approximation for counting primes. Such beautiful indirectness
The best part of this video is watching Dr. Grimes geek out over Ron Graham. :D
The animated sketches of Dr A -- and the 1970s typography are amazingly, disturbingly on-point.
"Numbah"
Peter Bergmann let me just clue you in, James is English, this is how words are supposed to sound
Peter Bergmann The English butcher their own language.
He talks weird though
Well you wouldn't pronounce it NUMBEARR would you
It's called an accent. The U.K has a lot of them. So does America.
Sometimes,I watch number phile even though I don't get anything because seeing the guy getting excited and enthusiastic about explaining, is somewhat fun.
please do a 70's Numberphile episode.
PLEEEASE!
In my mind, ln(x) is log_e(x), lg(x) is log_2(x) and log(x) defaults to log_10(x)
I know I just commented about this about 9 minutes ago, but it needs to be said again: *_JAMES GRIMES' ENTHUSIASM FOR MATHEMATICS IS JUST ABOUT THE PUREST THING ONLINE._*
P.S. (takes a long drag on a cigarette) Smoking is baaaaaaad.
It made me so excited that you talked about Isaac Asimov. He's my favourite person. You know that question "If you could spend an hour with one historical person who is no longer alive, who would it be and why?" I used to not know how to answer that question. But these days, I know it would be Isaac Asimov, without doubt.
Knowing that he wrote about this number has made me realize that I need to read more of his non-fiction stuff, like his science articles. I started reading a book of his articles once, at my school library, and I loved it! But for the most part, I've only been reading his short stories and novels. Time to scour the internet for his science article anthologies!
singingbanana in our hearth
+stefanilserbo sing along with him :D ;)
+stefanilserbo rest in peperoni
+stefanilserbo In our hearth? Did you light him on fire?
+Anonymous User I made a mathematical diagram of his body and then made a tattoo on my heart of his formula
stefanilserbo You said hearth, not heart, in your original comment
love the paisley in the background at 5:46 and the 70's bit at the start
I actually understood the formula he explained perfectly fine. The one where he said that "some of you won't be familiar with what I'm going to do." I feel like my studies have gone pretty well.
This is the Numberphile vid I've rewatched the most times. I always return to see the smoking '70s Grime, but then I remember what a cool concept is actually being discussed.
I am 15 years old. I watch numberphile.
I just can't get enough of James talking about numbahs :)
"We'd have massive ties and be constantly smoking" lol
The Really, Humungous, Gigantic, Enormous, Massive Skewes Number
1970s style numberphile go
3:50 Log(x) is normally x's logarithm--in base 10 in this case since no specific base was given (as far as I've been taught); ln(x) represents x's natural log and is in base e, of course...
That or some conventions have really evolved in the last 20 years...
8:48 : The use of an "x" as the multiplication symbol bothers me. It's even more unforgivable as it's written with a serif font and doesn't even look like a simple cross anymore.
My Dad likes to say that when he uses a word he does not know how to pronounce, it's because he's an avid science reader.
He's a dork but he's right!
You must sell the 70s Numberphile T-shirt, now, you know!
This is so amazing. I watch again and again. James is so wonderful as a teacher. He also says how it would be if Isaac Asimov talks about it on 70s Numeberphile. :-D He is superb this guy.
what about tree(3), are you also going to make a video about that?
Are you the guy on the Redstoner server?
Are you the guy on the Redstoner server?
Are you the guy on the Redstoner server?
is that a log joke then you should be ashamed of yourself
+Raihan Muhammad no pun intended, I don't even see it.
I love how I don’t understand this but still watch it.
We love James :)
Mathematicians: This inequality never flips
The inequality: I'm about to do what's known as a pro gamer move.
Whew. That sure takes a load off my mind. I'm always nervous that my inequality sign will flip when I'm dealing with powers of a thousand.
Lets get this guy to skewes' number subscribers.
But is there a number where the two functions give the same value? It's quite unlikely but possible...
+Flandre Scarlet I believe both estimation functions are undefined for x=1 and x=0
+Flandre Scarlet yes but not at an integer value of x
+Flandre Scarlet I may be wrong but since both functions are continuous for large numbers, the intermediate value theorem should be usable to show that there should be one specific number (exactly where the equality sign flips) that gives equal values for both functions. I imagine it would be a number with a long if not infinite tail of decimal places.
In short, as x increases, the functions have to get closer and closer together before the equality sign flips, very briefly becoming equal as the flip happens.
***** pi(x) isnt continuous at primes but li(x) is almost certainly never an integer when x is an integer
Thanks for the info :) Just to be clear I was referring to the approximation pi(x) = x/ln(x) which is continuous for x < 1. If there are points for which pi(x) < li(x) and for which pi(x) > li(x) then there has to be a real number in between where pi(x) = li(x) for a specific real x. That is what I assumed was being asked.
In case anyone else was confused round 8:00, no, I'm fairly sure the co-writer of How I Met Your Mother didn't co-write "A new bound for the smallest x with pi(x) < li(x)".
Wait, wait, wait now. You skipped something important. You claim that we know a run of integers where this inequality is flipped, but we don't know the first time it flips. So that means we somehow know how many prime numbers there are under some numbers, but not under smaller numbers. This requires an explanation :p. Please explain.
I think the issue is that the way we calculate pi(x) for large x is by using li(x) and the known error formula which involves related functions of li(x). And calculating li(x) for large x is not an easy task. This approach was formulated by Reimann and is greatly connected to his zeta function, so much of the discussion of the solution to this problem revolves around assumption of the Reimann hypothesis.
would you reply me after 4 years
This video sums up why i love this channel. So pointless and brilliant!
Edit: Also, I don't know his name but this guy talking is my favourite.
have you figured it out yet?
he's james grime
Question is how many times does it flip within Graham's number??
+Mark Lagunzad 42
+Mark Lagunzad A fair few
+ikschrijflangenamen haha i bet it's way way greater than 42
+Rhizopuz Stolonifer no.
+ikschrijflangenamen I would say 8
Big fan of Isaac Asimov; I definitely recall reading the essay JG is referring to. IA described it as the largest number usefully applied to a proof at that time.
I always thought ln(x) is the natural logarithm, in base e, and log(x) is the logarithm in base 10. Oh well, different notations again, i suppose.
+Cristi Neagu calculators, i.e. common CASIO fx-991ES has it like you know it too. I do also know / use it like it.
+Cristi Neagu Yeah the notations of logarithms are weird. ln(x) is usually the natural logarithm to base e, never seen something else. But then there is lg(x) which is either base 2 or 10, lb(x) and ld(x) usually base 2 and log(x) really depends on the context. Its often base 10, but it can be anything.
+Cristi Neagu In advanced mathematics, natural logarithm is conventionally detonated as "log(x)", since there is absolutely no need to use logarithm in base 10. If you ever need to use log base 10 (which you will probably very rarely do in advanced mathematics), you may just write: (logx)/(log10) (this is due to base change formula).
Conversely, In biology, astronomy, or engineering, natural log is almost absent and therefore log(x) will indeed refer to base 10 log.
+KevinJRattman For exactly the same reason, we almost never wrote "log" in my engineering studies (since there's no need for logarithms in base 10), but used "ln" for almost everything. "log" would mostly be used for the extended, complex-valued, version of the natural logarithm.
+Cristi Neagu In some contexts, ln(x) is denoted as log(x) where it's clear that the natural log is being used. For contexts where it's less clear, the notation ln(x) is used instead.
Numberphile: here’s a big number 😳
Here’s an even bigger number! 🤯
Does this mean that there is some real number x where π(x) = Li(x)?
The better way of saying what Greg said is that those functions aren't continuous, they're discrete, since you can only plug whole numbers into them. So no, there isn't necessarily a point where they cross.
+Brendan Beaver Functions that are defined on discrete spaces are necessarily continuous.
It's a basic topological concept, since every subset of a discrete topological space is open.
A discrete function is by definition not continous.
At the flip the functions will be close. Why do you think that they could not be equal there?
+hpekristiansen because they might truly have very close values, but we cannot say the values would be exactly the same at a certain point merely because the signs flip, since we do not know if the functions are continuous. Check out the intermediate theorem.
Seeing Dr James Grime = Instant Thumb up at 0:00.
Skewes lived from 1899 to 1988? That's skewed!
I always wonder, does he have all the numbers written down somewhere on paper outside the camera or does he just know these numbers by heart? He seems like such nice guy! Great video as always!
Wait, is this where the word 'skewed' came about in common speech as well, or is that a coincidence?
It’s a coincidence.
Source: en.wiktionary.org/wiki/skew
+cortster12 It's because Skewes got skewered once during a lecture and did a big number on it.
9:19 and it APPEARS that it’s gonna HOLD FOREVER AND THEN… it flips 😂😂
Have they changed nomenclature? I learnt that, if we are writing numbers on base 10, log means log on base 10, not on base e as they use on the video.
Log on base e is normally written as ln.
+Víktor Bautista i Roca once you finish high school, log becomes natural log. no one uses log10
That's a typical notation in high school. In university, they usually assume log is a natural log unless otherwise labeled.
+Rylan Edlin Except on calculators.
+ben1996123 In pure mathematics that may be true, but in science and engineering ln() is commonly used to denote natural log, where otherwise one would have to write either log (subscript) e or log (subscript) 10 to avoid ambiguity. (In science and engineering, clear, accurate communication is essential, and every item of nomenclature must be defined where used. In this arena some forms of shorthand notation have become universally adopted for the convenience of all concerned.)
+Víktor Bautista i Roca log is the traditional way to write natural log. Base 10 logs are easier to teach, so you learn those first, and we just decided to switch the usage of log to log10 for education. Now it's kind of muddled and recommended that you specify somewhere which is which.
So good to see James again!
There's a little bit of fudging going on here. James only uses the integer result from the Li(x) function and says that Pi(x) appears to always be less than. Under those conditions the rule actually fails for Pi(13):
Pi(x) = Int(Li(x)) (=6)
However, while James says "always less than" he actually contradicts himself by writing "less than or equal". Is he trying to have his pie and eat it, I wonder? Either way the pie looks a little bit sloppy ;-)
I love that paper, so elegant and simple even a kid could understand that.
I'm googolplexed by it.
* _ * ahh...
Ive been watching numberphile for over an hour now and this episode finally made me say it... I dont understand anything but i want to watch more
How could Stanley Skewes tell that the inequality flipped? I mean the Pi-Prime function has no closed form to this day, hence the notion for approximation right? So for recognizing the flip you would have to calculate and count all the primes of 10^10^10^34 by foot (or computer). But that doesn't sound like a task that could be done by computers yet, or at least by the computers of 50-100 years ago when he did this work. Do we have any genius here to resolve this question? ;)
+Patrick Fame you dont need to know the values of pi(x) and li(x) to show that one is larger than the other
Why not? How can you show one is bigger than the other without having actual values?
Patrick Fame because analytic number theory is magic
ben1996123
Not sure if trolling or just not eager to be helpful. I would really like to know. As a computer scientist, analysis and co are not my prime fields of mathematics.
Patrick Fame neither. i told you. analytic number theory. but don't expect to understand it unless you do a phd or something. the largest value of pi(x) known is only pi(10^26) which i think took about 15 cpu years and 128gb of ram to compute.
Remember as a kid thinking that a billion is the largest 😂
1 000 000 000 is a milliard ;)
+Laatikkomafia In english big numbers aren't called the same as they are in your language (and mine as well). So he's not wrong, but I get how it can screw up your head.
There's a Numberphile video about that.
+tabularasa0606 do you have a link to that? or do you know the name?
+Laatikkomafia billion
+xGhostModex "how big is a trillion?" is the name.
Love how excited he gets about the idea of a 70s numberphile.
The same guy made Tec-9 | Isaac and Awp | Asimov
Isaac Asimov? coincidence?
But is it stattrak?
Dayummm
Gauss did it it when he was 15 ! and I am already 16 and still at high school. I am hats off amazed and saddened at the same time.
How can someone be so great! Is it there surrounding or their unique enthu or their natural intellect or is it something else?
Natural intellect; he was very gifted :P
I can conceive an even greater number than Graham's Number. Vanhouten's Number = Graham's Number + 1.
vanhouten64 and we also know its last digit......It's 8 :)
Vanhouten's number plus 99999999999999999999999999999999999999999999
Infinity+1!!! (Saying it loudly makes me more right)
My number: The biggest number that will ever be found + 1. If you find a greater number than my number, it is still gonna be one bigger than yours. Checkmate.
Everyone can conceive a greater number than any number, but we're talking numbers that have been used in mathematical proofs.
Skewes Number is very, very small compared to Graham's number nowadays, like it was back in the 19th century when people thought no one could comprehend this number. But in reality there are many big numbers now so Dr James explaining this must have been surreal.
I can make a bigger number
10^10^10^35
kapa
noone cares if you cant use that number for a proof
+flawlessgenius but you care enough to make a comment about not caring.
+KillzGaming i care if you have a proof
that would be really interesting if you have a use for a number that big
+flawlessgenius boasting that this is a bigger number
You can always just add one, absolutely no one cares if you can create a larger number as anyone can.
Made me think of massive stars collapsing on themselves, shrinking in volume, passing that "flip" boundary in terms of density and then basically breaking fundamental laws of the universe.
So my number is 42^69^420^(number of views of this video)
I use this number in my theory about estimated number of atoms in the universe. Cheers.
the number of atoms in the universe is 10^80
João Victor Pacífico Well, according to my theory its square root of my number so... You are wrong.
João Victor Pacífico If you want to be a geek, at least be right. 10^80 is the APPROXIMATE number of atoms in the VISIBLE universe.
João Victor Pacifico That's a pretty bold statement, it's actually just an approximation
if this video has X views, 42^69^420^X ~ 10^(1.6*69^420^X) ~ 10^69^420^X (multiplying something larger than 10^10^100 by 1.6...) ~ 10^10^(1.85*420^X) ~ 10^10^(1.85*10^(2.6X)) ~ 10^10^10^(2.6X+0.25) ~ 10^10^10^(2.6X) ~ Skewes' number when this video has 13 views, and 10^10^(2.21*10^1985999) as of when this comment was posted.
when this video had zero views you get 42^69^420^0 = 42^69^1 = 42^69 ~ 1.01*10^112
6:22 UNTIL....oh...until (in very mysterious way)...that anticipation is killing me :O this is almost like a theatre!
Finally I finally get why physicists are testing the equivalence principle so much.
At 6:50, James says that the inequality sign flips infinitely many times. Does that mean that there is a direct relation between Li(x) and pi (x)? Meaning it's not just by chance that Li (x) delivers a value close to pi (x)? If the relation was accidental, wouldn't it verge off at some point? Can we learn something about prime numbers from Li (x) if we understand this relation?
I've come up with a method for estimating the number of primes less than a number that gives perfectly accurate results (though it is a bit computationally intense).
For the number of primes less than the integer n, starting with 2 compute the prime factors of the number and there are any factors besides the number itself and 1 then decrement your count (which starts at n) by 1. When you reach your number n your count should equal the number of primes less than it.
Though it is correct, this is not an estimation because it gives you exactly pi(x).
It's official, Dr. Grimes needs a perm.
Skewes' number is basically 10 to the power of 1 with 10 decillion zeroes after it. To better explain how massive this is, googolplex is 10 to the power of 1 with 100 zeroes after it. That means googolplex is 10 to the power of 1 with 100 nonillion zeroes after it times smaller than Skewes' number. To calculate that difference, the quotient of Skewes' number to googolplex is equal to 10 to the power of googol to the power of 1 nonillion.
I Am also 15 years old and i have nothing better to do than watching standupmaths and numberphile :D
This man has such passion. love watching him rant :-)
Understood the formula. Felt so smart.
I like the way Matt is like "you may not understand the maths I'm about to do"...... Pretty sure he could say that on half of his vids
I started out being mildly curious about Skewes' Number, but this quickly turned to intensely wanting to see a 1970s Numberphile episode starring a chain-smoking James Grimes with a super-wide tie and epic sideburns
Very nice cartoon of james on the thumbnail. I appreciate
James is downright Awesome! :D
Makes it gripping, always..