See all three videos in this series - Grant's Prime Pattern Trilogy: bit.ly/PrimePatternTrilogy

@defenderofwisdom

Жыл бұрын

I KEEP TELLING YOU TO FIND THE ALLSPARK BEFORE PRIME DOES!!!

@Peter-kn9jj

Жыл бұрын

very interesting

@scarletevans4474

Жыл бұрын

So... at 17:45, You say that is never converges to any number, as it wanders around and never approaches any number? But why? Are there multiple accumulation points? Why it doesn't converge?? Then, (at 18:35) after talking for most of the video about these powers of primes summed together with primes themselves, we instead just jump straight to something else, just reciprocals of primes (so without these powers?? Were these only for that "intuition"? What was the purpose to talk about them for so long?), ignoring what we were doing for most of the video, to "measure that in slightly different way" and we get that it... suddenly converges to 0.9959..? But didn't Euler proved that sum of reciprocals of primes diverge to infinity? What exactly are we doing here? Are we taking these reciprocals with different signs, depending on the team that they belong to? Why it was just "hand-waved" and neglected, wasn't this supposed to be the main conclusion from the video? What exactly converges to that number: 0.9959.. ? This all was slightly chaotic, with the last fact being just barely "mentioned" in a matter of seconds at the end of the video, without much of explanation, so I would be glad, if someone could add up to what he said and explain it slightly better or fill in the holes (like that divergence at 17:45) :-) Thanks!

@tbraghavendran

Жыл бұрын

Why isn't this video not available in 3blue1brown ?

@johnjeffreys6440

6 ай бұрын

i don't think numbers should be called, 'prime' because it implies the others are not prime. How about indivisble or non-composite?

"2 is disquialified. It's the ODDEST prime as it's the only one that is EVEN". I really love that statement 🤩

@tolkienfan1972

Жыл бұрын

3 is also the oddest prime in that it's the only prime divisible by 3. AND it's actually odd. Also 5 is the oddest prime, in that it's the only prime divisible by 5. Oh, but we have a name for divisibility by 2, so it appears more special, when really, all the primes have the same generalized property

@Tankem

Жыл бұрын

@@tolkienfan1972 I feel like you missed the joke there

@tolkienfan1972

Жыл бұрын

@@Tankem nope. I got it. My comment is an aside

@efhiii

Жыл бұрын

​@@tolkienfan1972 There's only one instance of two consecutive primes though; 2 followed by 3. There are multiple instances, possibly infinitely many, of two primes being separated by two, known as twin primes. This, among other things, make 2 particularly unique. Multiples of two being easy to identify & think about certainly helps, but isn't the only thing that makes 2 odd.

@argoneum

Жыл бұрын

Just a thought: in base 12 divisibility by 3 would also have a name, and corresponding last digits: 0, 3, 6, 9.

Grant is so good at pulling abstract concepts down to earth. He's always asking the question "Ok, so this is definitely true, but how could I have thought about it to have guessed that before proving it?" It really gets at the heart of what makes math so fun!

There are very large stretches of primes where 1mod4 is in the lead: 17,624,031 primes in a row from prime 18,536,693,261 to prime 18,953,583,223 87,113,527 primes in a row from prime 1,491,324,485,141 to prime 1,493,766,257,567 89,473,585 primes in a row from prime 1,488,610,998,109 to prime 1,491,118,967,167 But then there is also a very long stretch from 19 billion (19,033,524,533) to 1488 billion (1,488,478,427,089) where 1mod4 is never in the lead. Edit: Found a much longer stretch of 1mod4 in the lead: 2,998,321,421 consecutive primes from prime 9,193,124,324,353 to prime 9,282,636,959,627

@samp-w7439

Жыл бұрын

How do we know this?

@lasagnahog7695

Жыл бұрын

Thanks for sharing. I was wondering what the streaks looked like at somewhat higher numbers.

@tonygruff

Жыл бұрын

This suggests a conjecture about whether for any natural k, there is a stretch of length k where team 1 has the lead

@yoyoyogames9527

Жыл бұрын

​@@samp-w7439 presumably someone has checked using a computer

@yoyoyogames9527

Жыл бұрын

​@@tonygruff maybe not for every natural number but i wonder if they get arbitrarily long

0:40 "It's the oddest prime. [...] It's the only one that is even." What a peculiar sentence.

@jonasba2764

Жыл бұрын

Haha that's a classic joke number theorists make. It really is quite odd to be even as a prime!

@LemoUtan

Жыл бұрын

@@jonasba2764 Technically, maybe it'd just be easier to define numbers divisibly by two as odd and otherwise as even? Linguistically, the alternative - to redefine the adjective 'odd' as unremarkable and 'even' as bumpy - would only upset all English speakers, forever. It'd have to go to the UN or something.

@jonasba2764

Жыл бұрын

@@LemoUtan Hmm, so then 2 would be the only odd prime, making it linguistically odd as well. I like your idea, now we just gotta convince everyone to make the switch :)

@MenachemASalomon

Жыл бұрын

@@jonasba2764 Related is the theorem that there are no boring numbers. Because some numbers are interesting, like 1, or 2, and the first number to be boring instead of interesting - well, that's interesting, isn't it?

@LemoUtan

Жыл бұрын

@@jonasba2764 Yup. Shouldn't take too long.

Grant really has something about him. He's the type of guy you want on your team.

Жыл бұрын

Yeah, he has such a positive vibe, and he's amazing at explaining and analyzing things. I cannot imagine a single situation where a personality like this would be detrimental.

We need a 3B1B video on why that natural log trick works

@AssemblyWizard

Жыл бұрын

He also showed this in his 7th lockdown-math video (at 7:12), but also with no explanation

@trobin

Жыл бұрын

i just spoke to him and he said he would like to use that proof (which is apparently a pretty simple manipulation) to segue into a video about the prime number theorem and the riemann zeta function :)

@samp-w7439

Жыл бұрын

@@trobin What an insane flex - I just spoke to Grant Sanderson

@Alex_Deam

Жыл бұрын

A multiplicative series (like this one) can be factored in a clever way using prime numbers. The magic phrase you want to google for that is 'Euler product'. And log works nicely with products since log(ab)=log(a)+log(b). Then you write the logs themselves as a power series and the new series 3B1B gave pops out.

@toasteduranium

Жыл бұрын

Let me know when anything related to this comes out

I just have to say this before the chat becomes flooded with comments but I love numberphile videos. They inspire my interest in maths and now I am in national training. Thank you numberphile.

@numberphile

Жыл бұрын

You're welcome - thanks for the kind words

@benjaminshatokhin4725

Жыл бұрын

@@numberphile Thank you. You cannot believe how much it means to me.

@motherisape

Жыл бұрын

@@benjaminshatokhin4725 what's this log of series called I want to learn more about it . I want it's proof . If someone knows please tell me

@ssaamil

Жыл бұрын

same, I really enjoy Numberphile :)

@thej3799

Жыл бұрын

@@numberphile 💯

As a team 1 prime, I can confirm that we do live in a team 3 dictatorship, and that the composites are ready to help us revolt after grant mistreating them.

@robertpearce8394

Жыл бұрын

Perhaps you could vote to leave.

@alihms

Жыл бұрын

​@@robertpearce8394 Primexit?

@zubiprime

Жыл бұрын

@@robertpearce8394 We tried that a few years back, they blocked all the exits and still are blocking them.

@gansogames4927

Жыл бұрын

@@robertpearce8394 It's a prime time to leave.

@josephpazar

Жыл бұрын

😂😂😂

Thank you for clarifying the meaning of "almost" in this case, Grant. Probably and statistics have "almost certain" as a concept, which basically means certain. My own control field has a concept "almost strictly positive real", which are systems that can be made into SPR systems, where the strictly just means "not including zero" and the positive real mostly means "invertible". None of these are like the "almost" that humans mean, its fuzziness is a feature, outside of math.

@idontwantahandlethough

Жыл бұрын

Ok so I don't know if you meant to say "Probably" instead of 'Probability' in your comment, but it made me laugh. Thank you, I really needed that today

Grant Sanderson's (aka 3Blue1Brown) voice is just so soothing. I can listen to whatever this guy has to say.

@liliwheeler2204

Жыл бұрын

Colab between 3b1b and Hannah Fry? 👀👂

Also, ln(π/4) is negative, which corresponds to the 3's, so that gives the sense that when you tend to infinity, the 3's are the ultimate winner, even if they occasionally lose the lead.

@landsgevaer

Жыл бұрын

Hmm, but that constant is tiny if not negligible compared to all the terms that have even powers of primes...

@landsgevaer

Жыл бұрын

@@josephs.webster6484 Sure, I am just a bit surprised that the OP attributes the win to one term, out of an infinity of them. That requires a better motivation imho.

@anonymoususer2756

5 ай бұрын

But that’s from adding and subtracting the actual values of the reciprocals of the primes, not from adding or subtracting 1 for each prime.

"Do you know someone who's good at making mathematical visuals?" - love it

I have no joke been waiting for this video for 3 and a half months. So excited to watch this later today after rewatching the first 2 vids

It surprises me sometimes when 2 is singled out as a prime. Understandably, our definition of even vs odd is etched in our minds, but is it just because we gave it a name. We just separated the numbers into 2 buckets: 0mod2 and 1mod2. Or maybe we just defined the buckets as numbers with a factor of 2 and those without. If we had a name for numbers with factors of 3, would 3 be an odd prime as well?

@galoomba5559

Жыл бұрын

Yeah, sometimes you look at primes mod 6, and you have to single out both 2 and 3 as the special cases

@antosha4812

Жыл бұрын

I think it's also because 2 is small. In this case it needs to get singled out because 2 is the only prime that's 2 mod 4 since all numbers that are 2 mod 4 greater than 2 will have 2 as a divisor and so will not be prime.

@finngardiner5358

Жыл бұрын

I think what makes it feel special intuitively is that n mod 2 = 0 and n mod 2 != 0 are the same size, which is a unique property. Exactly how meaningful that is I don't know but I think it's what we latch on to

@Brainth1780

Жыл бұрын

Another reason 2 is "odd" is that because it's so low, there are no natural numbers lesser than 2 that *aren't* divisors of it. When we mentally check whether a certain number is prime, we go over the numbers up to it making sure none of them divide our number. But the only numbers up to 2 are 1 and 2, both of which *do* divide 2 evenly, so it feels like it's only a prime number because there's no numbers to disqualify it with.

@TheEternalVortex42

Жыл бұрын

2 has a lot of other annoying properties. For example F_2 (the field with 2 elements) is a counterexample to pretty much everything

I loved watching numberphile back in my undergrad days. Even though I studied neuroscience, learning about topics in math such as imaginary numbers through this channel really helped strengthen my logic. Glad you guys are back!

@5Xum

Жыл бұрын

Back? They never left!

I've been waiting since November for this video, and whether it's true or not, I'm blaming Grant's amazing visuals for slowing down the production process. What a great collab. Thank you both.

I find it interesting that not only will team 1 always win at least some of the time forever, it will always win by larger and larger amounts as you keep looking to higher numbers in order to keep the value at 0.005%. So eventually team 1 will lead for 20 billion primes in a row and that 20 billion will still only be 0.005%

@viliml2763

Жыл бұрын

Not necessarily. It could lead for 5 primes in a row then lose the lead for 2 primes in a row then take the lead again and repeat that 20 billion times

After months of waiting, right when I already start to believe the part 3 does not actually exist, here it comes ...

YES, FINALLY! I waited so long for this.

I've been waiting for a new upload between NP and Grant for way too long! I'm glad it happened!!!! Woohooooooo

chi(n) oscillates between 0, +1, 0, -1, so another way of writing it is sin(n pi/2). So sum( sin(n pi/2) / n ) tends to pi/4. Now, I asked myself what the continuous version of this sum would be. Turns out int( sin(n pi/2) / n ) over the positive reals is pi/2. Twice as much!

@thej3799

Жыл бұрын

Cuz in our 3d space here.... xyz, right? The sections these planes create if considering the 0 axis coordinate, sphere, or the reason fir circle in 2d, the outer bound of sphere only need radian. And that radian also defines perfect circle, at any relative angle. It's the meridian of sphere

11:16 One small thing to note is the "logarithm" in this sense only works for a completely multiplicative function, like χ(n) or 1/n^2.

@welcomeblack

Жыл бұрын

Is there a name for this type of logarithm, so I can go look it up later?

@thenonsequitur

Жыл бұрын

I think he did cover that a bit earlier, at 7:23

These are fascinating videos. I don't consider myself a math person but I find these videos are just as compelling as an engrossing novel you just can't put down or a mesmerizing painting you can't stop staring at.

Hey man been watching for years, and thank you so much for all the knowledge you provide to us. It's truly inspiring.

A collaboration between 2 of the most valuable channels on yt

@oresteszoupanos

Жыл бұрын

I like that prime number you snuck in there 😀

Gotta love these two during Uni times. They were better lecturers and conceptualized the ideas better than most professors.

@well_said7846

Жыл бұрын

Essence of Calculus was easily better than even Gilbert Strang I'd say.

@theblinkingbrownie4654

11 ай бұрын

​@@well_said7846id respectfully disagree

Just listening to Grant speaking is also soothing

I think I'm in love with 3B1B ! You're amazing, Grant! Keep it up!

Hearing Grant saying susses makes my life complete

Great to see Grant again!~

Crucial point: ln(pi/4) is negative (since pi/4 < 1). That's the intuition for why there are more 3 mod 4 primes than 1 mod 4; even after putting all the even powers in the positive bucket when half of them "should" have been in the negative bucket, the total sum is still negative.

@benp2291

Жыл бұрын

It doesn’t matter what the value of ln(π/4) is, so long as the series in question converges. The idea grant spoke about is that the sum of the positives go to infinity, and the sum of the negatives go to -infinity; however, added together they converge to a value. This is the big point - for every positive value we need a negative to stop the sum going to infinity. Why should there be more 3 mod 4? Well that’s because there are more positives as you get lots more from prime powers, which means you’re left with basically only the primes that are 3 mod 4 doing the negative business. Which means more primes 3 mod 4 to stop the sum going to infinity. This is overly verbose but I hope it explains that you don’t care what the sum converges to, just that it does and we have the positives sum to infinity and the negatives sum to -infinity

@pifdemestre7066

Жыл бұрын

@@benp2291 The only existence here of the sum is not enough in the argument, the value is also important. The sum of reciprocal of squares of prime also converge, so for the team 3 lead intuitive 'argument' to work what matters is the fact that essentially: sum of reciprocal of squares of primes (+ some more) > ln(pi/4) + sum of reciprocal of (some) cubes of primes (+ some more) + the 'some' can be computed by looking at the chi function.

@kwcy92

Жыл бұрын

We’re talking about natural log here, so ln(π/4) is negative ∵ π/4 < e, even though π/4 < 1 is true.

i thought this was never coming😭😭..i assumed ur footage was corrupted or something..so glad to see it finally posted ♥♥

When these 2 do a collab, you know it’s gonna be good

@Robostate

Жыл бұрын

You mean, when these oddest primes do a collab!

It was fun to watch Grant tripping around on the word 'almost'. Mathematicians' language really is different than normal people.

Loved that, and love Grant's work. He is one of the best explainers of difficult stuff I have ever come across on YT.

This is the most bizarre mathematical result I’ve heard of in a while.

Love you guys doing these videos.

Always good to see Grant on numberplate 👍

The log "shortcut" reminds me of leap seconds. Basically just some way to include some terms to get closer to a desired result.

This is probably the most involved and difficult to grasp of Grant's videos ever, I'm gonna need multiple rewatches

For some reason, I chuckled when he said "2 is the oddest prime"

@thej3799

Жыл бұрын

5 is. Because it's not half base 10

Grant is great as always!

My initial thought was well, any even power of an odd number will be of the form 4n+1, so since primes occur in the first place they can kind of by definition, there's less "room" for primes of the form 4n+1 than for 4n+3

4:45 my simplest explanation for why is that prime squares can be 1 mod 4, but not 3 mod 4 that accounts for the difference since powers of primes are considered to be partially prime for theorems regarding the prime number theorem (see the Van Mangoldt function for example)

Another fun and interesting video, thanks!!

I am curious about the "distance" away from 0 for Team3-Team1. Is that bounded? Or does it increase for both and just balances out?

I'm going back and forth on the fractional odds of getting a quarter of pi.

Let's go another Collab btw 2 of my favorite math channels

We love the crossover and collab!!!!!!

Just a few days ago I watched the first two and was wondering where part 3 was. What a great payoff.

I love this guy, the best math tutor

I just want to say that, your videos always be able regain my interest in math when I feel burnout, it also help me to see problems from different perspective which is important for a math lover Wish you guys all the best!!

There's some beauty in this pattern!

Always great to have Grant teaming up with Brady. I gotta say though, as a number theorist myself, the "weirdest possible way to take a logarithm" does such an injustice to the actual process happening behind the scenes. In reality, it's a completely natural way to express a logarithm. Granted, you have to understand the notion of a completely multiplicative function, an Euler product, and Taylor series, but I don't think these are concepts that would be challenging to the typical numberphile viewer. At the very least, the rigorous calculation (or enough details to demonstrate it) should be included for those familiar with these topics. That critique aside, I really liked the intuition given here.

@ps.2

2 ай бұрын

"Completely natural way to express a logarithm" - I see what you did there.

@joshuastucky

2 ай бұрын

@@ps.2 😂 didn't even catch that when I wrote the comment.

My first intuition was: 2 numbers multiplied that are both 1 mod 4, result in another 1 mod 4 number. 2 numbers multiplied that are both 3 mod 4, result in a number 1 mod 4. Only if a 1 mod 4 and a 3 mod 4 number is multiplied, we get a 3 mod 4. So I thought if we think of the Sieve of Eratosthenes, there would be a higher chance that 1 mod 4 numbers get eliminated. But didn't think this through further...

@quentind1924

Жыл бұрын

That does not mean anything. 1 number being 1 mod 4 and the other 3 mod 4 is as likely than having both 1 mod 4 or both 3 mod 4 (if you flip a coin, you have 50% to not get the same result) You have 4 possivilities : both number you multiply are 1 mod 4 (the result is 1 mod 4), the first one is 1 mod 4 and the 2nd one is 3 mod 4 (the result is 3 mod 4), the first one is 3 mod 4 and the 2nd one is 1 mod 4 (the result is 3 mod 4) and both are 3 mod 4 (the result is 1 mod 4). So half of the time it will be 1 mod 4 and half of the time it will be 3 mod 4

@snowmaninthepool2644

Жыл бұрын

​@@quentind1924 well, yes but no. My first post is a bit misleading. The three sets don't have 1/3 chance each. But its also not a coin flip here. Lets say we have m primes 1 mod 4 already in the list, and n primes 3 mod 4. Then there are m(m-1)/2 + m = m(m+1)/2 combinations of 1 mod 4 primes ( m(m-1)/2 pairs different numbers and then there are m self-combinations (square of the primes) Same logic there are n(n+1)/2 combinations of 3 mod 4 primes Last, there are m*n combinations of different primes. m(m+1)/2+n(n+1)/2>(m^2)/2+(n^2)/2>=m*n So there are always more pairs that eliminate 1 mod 4 numbers than pairs that eliminate 3 mod 4 numbers.

The wait is over! And it was worth it :)

Two questions: 1. What is the LARGEST LEAD "Team 3" has? Is there an upper limit? 2. Who the heck comes up with these things? "Gee, I wonder what it would look like if you did this weird thing with prime numbers...."

3B1B! Love this guy as a guest and his channel! Edit: this was somehow inspirational. Great video 👍

Fascinating video! I don't know if it's already been mentioned somewhere in the comments, but the fact that ln(π/4) < 0 lends even more credence to the theory being true on principle alone, as it requires even more tiny negative fractions to fill the bucket and tip the scales that direction.

9:35 - What's shown on the screen is not "reducing it by one third", though. It's reducing it *_to_* one third (of its original value). Reducing it *_by_* one third would mean subtracting 1/3, and reducing it *_by_* one third of its original value would mean multiplying it by 2/3.

This was a much better waste of my time than many other videos on KZread. Thank you.

We are waiting for a great video on 3Blue1Brown after getting 5 million subscribers!🥰

Classical mathematician shortcut. Just remember this short, simple, absolutly uninuitive, 20 step instruction and then its totally easy to work out :-) Still the fact, that team 3 never gets in the lead is verry nice. Those problems nomaly have the tendency to go to 0 or 100% or infinety, or 50:50. Something you would expect. Thx for sharing this. great vid.

@ps.2

2 ай бұрын

Ha. Well, we think it look complicated, but that's because our normal logarithm algorithm is to hit a button on a calculator or look it up in a log table. The _actual_ other log algs are not so easy!

Brady's tortoise and hare comment at the end is interesting given that the two concepts meet in cryptography where you are modulo some prime and both its chi value and cycle length (found with Floyd or Brent's tortoise and hare method) are important.

I think another reason you should have some intuition that there are "more" 4n+3 primes is that (4n+3)*(4m+3) is always some number (4k+1). However, the converse is not true. (4n+1) * (4m+1) is also 1 more than a multiple of 4. There are just more ways to get one more than a multiple of 4 than three more by multiplying these numbers, and this allows 4n+3 primes to stay in the lead for longer as their products "take up" more space in the numbers 4n+1, and less composite numbers can break into 4n+3 territory.

Nice talk about Chebyshev’s bias with it mentioned against Dirichlet’s theorem of arithmetic progression while using Dirichlet’s series.

19:26 "No!" means "What have I just been explaining to you for five minutes my dude!" 😄

Worth the wait :D

That's a very interesting heuristic way of explaining the bias. Still though, all leads are temporary, and the asymptotic density of each of the residue classes is 1/2.

I did not think i would like this video that much. But as is always the case with numberphile, i really enjoyed it.

never heard of that method for taking logarithms :0

Grant is such a gem

This makes sports rivalries quite interesting, in a way. If you just showed me the graph and asked me who's "better" I would say team 3 without thinking, but if you asked me at some point in time and I only were to take current form and recent results into consideration, it'd practically be a coin flip. If a "season" started at the beginning of the series and ended at a random point, team 3 would be 99.5% likely to win the title. If it started at a random point and ended at a random point, team 1 would win about half the time. Either this is mind-boggling or I'm completely wrong. I really can't tell, because I'm mind-boggled.

Amazing channel. Ty

In one video, you manage to touch on the prime numbers, pi, e (via ln), and the most mind-boggling way to calculate a natural logarithm ever seen. mind blown

Brady's questions are on point.

Thoroughly interesting video.

Please make a video on Disarium numbers, and finding out the 20th possible disarium number 😀

Yeeees!!!.. been waiting for this one for sooooo long, lol

The way the difference was plotted as a function with integer values being positive and occasionally negative gives a really nice way to think about this. The question with the answer of roughly 99.5 % (if I understood it correctly, but I think I did) is equivalent to asking what part of its values are positive. If the ratio of positive values to negative values was 1:1, the 3s would be in the lead half of the time - but that's not the case. But it gives me an idea for another, similar question. How does the integral of this function behave? Both from 0 to some finite bound X, and on the infinite scale (0 to infinity) - that one I think is most probably diverging given the heavy bias towards positive values, but it's not impossible for it to be finite-valued given this fact (meaning that you can have a function that's positive 99.5 % of the time and yet has integral equal to 0 - the negative areas just have to be deep enough to compensate for their rarity).

@ps.2

2 ай бұрын

"Integral" of a discrete function? Do you mean a Σ sum, or can this be thought of as an actual integral in some [no pun intended] way? Anyway, I agree that it probably diverges, and the _lim n→∞_ is gonna be like -1/12.

I didn't quite notice we were talking about primes yet, and it took me a moment to work out why there were only two categories of number mod 4... :)

Being aware of what seems like a natural imbalance, have there been any deeper looks into the commonalities between the threshold points? Given the appearance of pi, plus the actual results data, it feels like there must be some periodic pattern on a spiral that guarantees a minority batch within some max interval, and whose duration is commensurate with magnitude of n. Anyway, I'm a first time commenter, but have watched and enjoyed both of your guys videos for a long time, keep up the great work!

I can't help but feel itchy by this missed chance, in the "teams" plot, to have made 3 "blue", 1 "brown" ...

Is there a common name for this logarithm technique for those who might want to look for more info?

Seriously, these are the kinds of friends I'd love to have. In my dream world, I live in a neighborhood where all my neighbors are guys like this - and their wives are doctors or something like that.

You know the video is gonna be good when it has Brady in it

Finally, the trilogy is complete

There is always hope!

I'm studying quadratic sieving at the moment. For factoring

Does this mean that for p>3 the sum of the reciprocals of the team 3 primes is the same as the sum of the reciprocals of the team 1 primes?, I.e. if you calculated the 'pseudo-alternating' sum 1/5-1/7-1/11+1/13+1/17-1/19-1/23 ... it'd converge to zero?

Play the same game flipping a coin for some large number of flips. Team H and team T. Whoever starts out leading, will tend to hold the lead for the duration of the flips. Both will have about the same number of wins, but at any given moment in time, whoever started out leading will have a higher number of wins. And this is random, unlike the primes race.

I find this topic analogous to entropy. How for most of the time everything is barreling towards chaos and disorder, we find these small bursts where order and complexity emerge, like in our own pocket of complexity we all live in.

you videos are responsible for a math teacher saying i "know a lot about primes"

3:32 I think it's more of a bias in our language: we have a word for numbers that are divisible by 2 (even) and another word for numbers that are not (odd). Prime numbers have also a bias towards throdd numbers (3 is the only threeven number), 5 is the only quinary number, ...

@markmontgomery2171

Жыл бұрын

Aww, I should have read this before commenting. Seems like I had heard the term "throdd" before but just forgot about it.

Also, ln(pi/4) is negative, which also tends to tell us, that team 3 should always be ahead by a little bit.

This was fun!

Grant did a video explaning the series converging to pi. It's an old 3B1B video

Finally the third part arrived :D

Finally, the final of the trilogy!!

@backwashjoe7864

Жыл бұрын

It’s been so long that I forgot that there was a trilogy!