TREE(3) (extra footage) - Numberphile
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- เผยแพร่เมื่อ 19 พ.ค. 2024
- Main video: • The Enormous TREE(3) -...
Featuring Professor Tony Padilla.
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Don't you hate when you're trying to prove how big TREE(3) is with finite arithmetic, but then the universe resets itself.
reminds me of Hitchhikers guide to the galaxy. The answer is easy yes it is finite the proof is very long.
I totally hate it!
I was so close last time I tried. Oh well, maybe this time I'll have better luck
That happened to me Tree(3) times already.
"I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."
"This IQ test stumps most mathematicians! Finish the sequence 1, 3, ..."
I was just thinking about trolling my friends with 1,3...
RBuckminsterFuller many answer 5 or 9 or 11 or 18 or 29 or 78 or 722 or even asceding so >3
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According to the Online Encyclopedia of Integer Sequences, 4 is an acceptable answer
In the sequence is infinite you can't finish it...
Well now I want to know if TREE(3) is prime
You can assume it's prime for now since it doesn't have any known non trivial divisors :P
@@priyansh1210 That's a dangerous assumption ;)
wonder if its possible to calculate that probability
First try to prove that tree(3) is odd
The guy said the closest you can get to knowing anything abt the number is the number of signs needed to prove it s finite...
The class: Tree(1)
The homework: Tree(2)
The exam: Tree(3)
The test: tree(3)
The finals: tree(tree (3))
TREE(Infinity)
@@playmaker4700 Isn't that just infinity anyway?
The Job Interview: Tree(Tree(Tree...(3)))))))))...
@@keafoleafo8368 yes, any size of infinity (say omega) put into TREE should return infinity. I don't know if it would return the same size of infinity or not though
"I've discovered a remarkable proof of Tree(3) theorem but the universe is too small to contain it"
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fossilfighters101 "also my brain is too small to contain it"
What a shame we don't live in a quality universe that could fit tree(3)
Only acceptable place to actually use that excuse
In base TREE(3) it is 10.
A odgovor na prvo pitanje?
And in binary the first digit is 1.
Can you give us their alphabet here?
What a useful base that is
Subhransu Mohapatra not necessarily
"The universe will eventually reset itself."
"The universe will eventually reset itself."
Once comes around what do you feel, I love Jack woke up press and seal me big pain to Pono.
(speech to text, Not what I meant but too funny to not post)
"The universe will eventually reset itself assuming that that will happen forever and that the universe is a perpetual machine, otherwise eventually everything will end forever and space time will cease to exist."
What a happy thought to think about while you're alone in the house!
Looks like we had less time than we thought
Repetition legitimizes
Repetition legitimizes
11 11 It’s impossible to prove or disprove that it will. We can only make more and more assumptions.
Edit: or we can just accept one theory, which is fine, as none of us will ever live long enough to find out the validity of said theory.
And still, TREE(3) Is closer to 0 than infinity.
so is every cardinal
@@SoloLevellor except my ego
Massimo Del Bianco depends on which infinity
It’s faster than a function of Epsilon sub script zero
Infinity divided by 3 would be closer to zero than infinity. Well, it would also be infinity. Wait, what?!
We're gonna need a bigger universe.
If you were to increase the universe's size by a googolplex factorial ^^^^^ a googolplex factorial-fold, then tried to fit TREE(3) cubic Planck lengths in there...you couldn't do it.
I reckon we'll need exactly a Graham's Number of universes to write down Tree (3), assuming one digit per Planck unit. Call it intuition.
No, you aren't anywhere close.
kcthewanderer I’ll go to Costco and buy one, be back in tree(3) minutes
jawad mansoor I’ll have to remember to order one next time the universe resets
If numberphile has Pi as their picture.. Numberphile2 should have Tau as their picture.
PallyNut you right, you right
Yes!!
This needs more likes
tau rules, change my mind!
also, this needs way more likes :)
NumberphileTREE(3) for SERIOUS insiders.
What they teach you in class: Tree(3)
What they ask you in the exam: Tree(Tree3)
What they teach you in class: 1 & 3
What they ask you in the exam: Tree3
😨😨😱😱😭😭😭😭
My brain just collapsed Tree(3) times
Im scared this like 11!!!!!!!!!!!!!!!!!!
@@SystemOfATool
Class: 33
Exam: Tree(3)
That TREE(3) will be great for getting LOG(3)s!
bruh lol
also log(3) ¬ 0.477121
clever
Lol. The question is how many LOG(3)s does a TREE(3) give? You will need multiple axes to figure that one out.
@@georgesmyrnis1742 Likely millions of axes, if not even more than that.
But is it prime?
same question
Maybe there's a way to prove whether it's odd or even.
Dan Kranda almost definitely not, every time you go up and find a prime while trying to divide to see if it's prime, you add that number to you're division pool. Since tree(3) is sooo big you have so.... Many primes to divide by its almost definitely not prime. plus half of all numbers are instantly taken out by dividing by two.
Well the frequency of primes is like 1/ln(x) so I'd give it a 1/ln(TREE(3)) chance of being prime... aka 0
Well TREE(1) and TREE(2) are prime so it isn't unthinkable, but I'm gonna go out on a LIMB and say that it would be tricky to definitively prove either way
edit:
before I get called out, I totally forgot 1 isn't prime, but I couldn't resist the pun
Totally dissapointed, this video should’ve been called “(extra foliage)”
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I knooow!
Emilio Herrera "Lagniappe foliage"
(extra brown paper)
Well, at least we know that the entire universe is not just a simulation being run to calculate TREE(3) then.
Brilliant. My sense of free will is now secure!
@@tb-cg6vd Brilliant to See your Comment, But there is another Video here
Brome to See your Comment, But there is another Video here
@@tb-cg6vd keep in mind, its a "sense" of free will. Not free will itself ;)
Actually , it is. We are just the bootloader.
1,3, Visible universe collapses into a singularity
"Exponantiation on steroids"
scalpian your thing, to the power of TREE(TREE(TREE(3)))
ExponenTREEation!
Symbol juggling on meths.
Claudio Acsinte Exponentiation*
Don't you hate it when you're doing proof for your maths homework and the universe just resets itself....
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Az A
Omg yes
You said that last recurrence...
+uvuvwevwevwe onyetenyevwe ugwemubwem ossas MY BRUDA
Az A hold on lemme go buy is a new one at ikea
When he wrote Tree (Tree(3)) I got anxious because I thought the universe was going to crash.
"So it's never been done before?"
"Whoa-whoa-whoa-whoa there guy. Just hold your horses. The question is CAN it be done?"
LOL
the universe will eventually reset itself, the universe will eventually reset itself.
hah! well played
A J
Lol I scrolled down hoping someone else saw that haha
BoWeava They did the same on the poincare recurrence time vid
yes due to there only being a finite amount of states that the universe can be in. Even if some of the states are infinitely big.
CarBricksCity niiice, haven't seen that one
The thing I don’t quite get about poincare recurrence for the universe is that the recurrence theorem requires a sequence of sets that is bounded. For instance, gas molecules in a closed box is a bounded system and a sequence of states of those molecules within that box will repeat themselves according to the theorem. But the universe is expanded and therefore the system is unbounded so I’m not quite clear on why the Poincare recurrence theorem applies. To take the gas in a box analogy further, if the box is instead an inflating balloon and the balloon can inflate indefinitely then there is no guarantee the molecules will repeat states because they have paths available which can expand outward with their boundary. Similarly the particles in the universe can expand with the universe so it seems like there would be no guarantee their states would repeat (since part of their states includes their relative positions in an expanding spacetime.)
I’m not saying the video is wrong, I’m just confused how this is resolved for an expanding boundary.
You know what's even crazier?
TREE(3)^0 = 1
And 1/TREE(3) is really small.
Yeaaa, the real deal still is Zero, the number which demolishes everything else.
well ..... -2 is smaller.
Félix Pinchon
TREE( TREE(TREE(TREE(3))) )^0=1 too
Wtf universe
ah so the zeroth root of 1 is TREE(3)! We found the solution boys!
"Exponentiation on steroids" Best description of Arrow notation I ever heard.
I came across TREE (3) yesterday when I was watching an online documentary and it both blew my mind and excited me immensely. I'm not a mathematician, I'm a musician, but this is just so awesome. I love this guy's brains and enthusiasm. Anyway, we were looking for a name for our new band - so calling it TREE (3). I hope no-one else has that name, but I love this so much. Thanks! :)
I would name a band 6EQUJ5 and pronounce it "The WOW Signal" lol
I'm smart guy math what's the point I understand to try understand Googleplex the numbers so unimaginable at its but so what's the point Graham the numbers so unimaginable what's the poin going beyond t 😂😂
@@IsaacHarvison-mt5xtwhat
Is TREE(3) closer to TREE(2) or TREE(4)? Do we know anything about the growth characteristics of the TREE() function?
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TREE(n) is always going to be closer to TREE(n-1) than TREE(n+1) in terms of absolute size. considering TREE(4) is just TREE(3) + an extra seed , you could just write out TREE(3) and then repeat entire structures only changing the color of one seed, effectively nearly doubling the size. And that's just changing the color of the seeds using 3-seed structures already constructed, not counting all the entirely new trees you could make using all 4-seeds
Scot Brown TREE (3) is way closer to -TREE (3) than to TREE (4)
HopUpOutDaBed Why nearly doubling? I think, without consider the 4-colour trees, you'd already get 4(TREE(3)). Using RGBW, you could do a TREE(3) with RGB, RGW, RBW, and GBW each.
+HopUpOutDaBed - I like the way you think, that's a very elegant proof!
Well, TREE(3) is clearly smaller than the sum of all natural numbers, therefore, an the upper bound of TREE(3) is -1/12
bivtyfrcygvubugwerdcfuvgibjhvibobhjhb!
I can't take this!
Yebjic
Yeah thats something i dont get about infinity too
Only in Riemann Zeta function. Watch Mathologer video for full explanation. The one done in response to Numberphile video on -1/12.
We do have the upper bound for TREE(3)
It is clearly less than TREE(3)+1
@@vishalarya93 yes, welcome to the joke
Thanks to this channel I have fallen in love with math and I am really considering studying maths!
Did you study maths
I love how happy he was at the end describing his joy over this type of math.
Surely TREE(n) grows faster than LOG(n)
Balazs Lovenberg it sure does
Man that's an amazing comment , I wish I thought of it :)
You should also consider ROOT(n), because it grows slower than TREE(n) too.
Hard to tell but yes I think if we examine the growth the TREE function just edges it out.
I once heard of an infinite divergent sequence but later it got summed up to -1/12. You never know man. You. Never. Know...…...
It gives me a new appreciation of infinity.
But you're still not even close. lol
When he started nesting the Tree()’s, my nethers clenched fearing the universe might rend.
Oh, the universe reset itself again.
Man, I hate it when that happens.
no need to repeat, we can see itno need to repeat, we can see it
@@aasyjepale5210 haha, haha.
What if you filled the universe with mathematicians the size of a plank length and then they split up the work?
Probably my favorite part about 2017 was this comment because I just imagine a world of tiny scientists talking about numbers perpetually in the multiverse somewhere and that keeps me optimistic about life. I also would love to see what would happen if someone figured it out and the news spread across the trillions of tiny scientists like a wave of celebration as the universe rejoiced in finding the answer. Would it cease to exist since it's purpose would be fulfilled? Would the scientists find another problem to work on? Perhaps they would colonize different universes or even just their own ones and delegate the lesser scientists to act as the land masses. Neat.
yeah they not gonna get nowhere
Won't work either
@@axelpeneau2288 Yep.. Anything we can (reasonably) write as x*10^y notation won't even begin to tickle the things that require the double up-arrow notation, no matter how big y gets.
Where would they add the symbol?
"The universe is too small to contain it." I'll use this excuse next time I haven't done a due essay.
It is a tradition for me to come back to Graham's number and TREE(3) every once couple of years.
Spongebob: you know what’s -bigger- than tree(3)?
Patrick: what?
Spongebob: Tree(4)
And you know what function is faster and larger than TREE ? Subcubic Graph and Busy Beaver 😂
This should've been included in the original video!
I know a true believer like you would watch, but if you post a 19-minute video to TH-cam you may as well hang a big sign on it saying "DON'T WATCH THIS"
Better to post a video on the essentials, then a second video for people who want to go deep?
Numberphile2 why not a 3rd? Or maybe 4th! I surely won't mind :)
Numberphile tree (3)
You could have at least posted the pre-emptive TREE(TREE(3))
Thanks for mentioning the bell. Was wondering why I wasn't being notified. That said, what's the point of a subscription if not to notify you of new videos?
10:30 There's one contender to the TREE function that can absolutely batter it - SCG (Simple Subcubic Graphs). The problem is that I can't even begin to understand how and why that number is so big, so I guess my video request would be one on SCG.
Big FOOT
Utter Oblivion is bigger. Although I suppose you could just mention Cantor's idea of absolute infinity to end any big number discussion there and then.
Zaephou what would be far more interesting would be like if you found another number that was like less than TREE(3) orders of magnitude from TREE(3), like if it was actually coincidentally closeish
They used the same editing joke about the poincare repeat conjecture twice!
They used the same editing joke about the poincare repeat conjecture twice!
2:31 "you might remember what this arrow notation is... exponentiation on steroids" lol
And This is why mathematicians have more fun :)
They're just not bounded by the physical reality :)
I agree :)
Matt Parker should estimate TREE(3)
Markovisch He could, but he doesn't bother doing it.
At least he tried XD
It would be like a kid estimating the number of stars in the night sky.
"How many stars do you think there are?"
"Ten."
His answer would be a Parker Tree.
PARKER(3)=10
And 2yrs later, TREE(Graham's number) has been discussed
That escalated quickly
Soumyadeep Bhattacherjee well to be fair this video already goes way beyond that by talking about diagonalized recursive trees
TREE(Gaham's number) is less than TREE(TREE(3))
@@abombata If we assume the function grows with the input, and never drops (easy to prove) then your statement follows naturally from knowing that g(64) < TREE(3), so TREE(n) will be larger for the larger input.
And TREE(TREE(TREE(TREE(TREE(TREE(...TREE(3))))))) still doesn't match SSCG(3), even if you nest it TREE(3) layers deep.
Love the excitement of Tony while educating here, these massive numbers are just jaw-dropping from the explanation alone.
Have been waiting for this number since over a year
I've been waiting for it since the original graham's number video. When that video was uploaded i was hooked into big numbers and started checking all kinds of different bigger than graham's number numbers. Soon I met the king of them all tree(3) and have been waiting since for numberphile to do a video about it. I wonder if there are any bigger numbers that have been used in math (so obviously not arbitrary ones like tree(3) * 2)
hey ash
could you say you've been waiting for this number since over T(3) years?
@frizider2 look up SSCG(3), or even worse SCG(3).
SCG(13)
Shouldve gotten Prof Moriarty to do it so he can say "Tree Tree" over and over again.
I had the same thought! That would be quite enjoyable
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Lauraphoid hah would be pretty cute
"The universe resets itself - This is a disaster." Literally that is what disaster means, the disappearance of stars.
"dis" in disaster refers to unluckiness, not disappearance.
How about TREE(TREE(3))?
EDIT: damit, already done in video
I have discovered a truly remarkable proof that tree(3) is finite, which this universe is too small to contain...
fire eye exact words from Fermat if he is still alive today
lol
You have a weird concept of "discovering" something that categorically cannot be contained by your brain.
I expected you to use FOREST(n,m) instead of TREEm(n)!
Zejgar is that a factorial?
Might as well be, the cheeky fucker.
Can't see one for the other though
would have, but didn't see the FOREST for the TREEs...
Love the universe resetting itself editing joke
I watched both these videos, but I'm still curious HOW they know it's such a huge number.
given that I'm pretty sure the answer to that was someones dissertation, I'm not sure it would comfortably fit into a youtube video, lol
FYI: It's spelled KRUSKAL'S if you're interested in looking into it.
I know that the first digit of Tree(3) is 1
in binary
Skippy the Magnificent and in base TREE(3) the first digit is also a 1
Wow. Just now I realized that first digit of every number in binary is 1.
Like this is obvious but I never thought about it, thus only now I realized it.
@@coolguy4989 underrated comment
@@eliorahg explain 2
@@lunox8417 2 is "10" in binary.
Absolutely love this topic. I’ve watch this episode about x20 times over the last year and I smile every time.
Great work guys
You touched on the thing that fascinates me the most. Staying strictly with finite numbers, it's still the case that, no matter how you define a large number - TREE, iterated TREE, busy beaver, whatever, almost every number is larger than the number you've defined. Thinking of that fills me with wonder.
In fact proportionally, EVERY number is bigger
@@Amethyst_Friend Yes. If you select a random positive finite integer (yes, the concept of a "random integer" is problematic, but you know what I mean!), the probability of that integer being smaller than any defined integer (Rayo's number, whatever) is 0.
I'm curious if the size of Tree(n) increases with any kind of regularity as n gets larger. Like if you had an ungodly Cartesian graph where x = n and y = Tree(n), would there be some sort of recognizable pattern in, say, the first 100 y-values? Or does something crazy happen like Tree(57) isn't as large as it "should" be based on all the previous Trees?
I really want to know more about the growth of the Tree function. I don't really know how much progress has been made (or can be made) in analyzing it this way. After all, Tree(3) doesn't have an upper bound (aside from definitely being finite).
That is really interesting to think off, just like a logarithmic scale we need one for googological numbers like Graham's number and TREE(3) to visualize just how much bigger these numbers are!
Given that the TREE() function has a similar kind of rule set to the permutations of those objects (I am not a mathematician, mathematicians would probably strike me down for saying such a thing), then given that analogy they would probably do something similar in a way as each TREE(n) theoretically would 'contain' the lower TREE() sets within them plus all of the possible permutations of those sets with that extra seed color.
I wonder if this has anything to do with Group theory as I just realized I'm starting to pose a similar sort of question...
Sounds like you're asking if TREE is monotonic
Dr Tony Padilla, I would love if you talked about busy beavers! I mean, Tree(3) is big alright, but it's still a computable function. Big fan of your videos, really love your enthusiasm!
Shouldn't that be a computerphile video. n-state turing machines.
It's already on the Computerphile, and prof. Brailsford videos are one of the best ones there.
I'd love to see a proof that TREE(n) is a computable function. I'm not sure about that and I haven't seen a proof - although I've seen it being mentioned that it is computable several times.
@@isuller A function is computable if there is an algorithm that can (given enough time) compute it. The simplest proof that the Tree-function is computable would be an implementation of that algorithm - it doesn't even need to be very efficient. We can even do it a normal programming language. The naive algorithm that requires the least imagination would be to do an exhaustive search of all possible forests for the given n and return the number of trees in the largest legal forest. The trickiest part would probably be to do the test for inf-embedding - but still conceptually doable. Feel free to reply if there are any questions! :)
Just a note but this actually happened and he spoke about them in the video regarding Rayo's Number.
I wasn’t paying too much attention bc this was background noise to me kinda, but if TREE(3) is 2^^1000, the last digit is a 6. Assuming I’m doing this correctly, 2^^1000 = 4*2^^999 = 16*2^^998, etc. since 16 ends in a “6”, and any number ending with a “6” squared results in a number ending in a ”6”, BOOM! You have one of the digits you need. Progress has been made.
2^^1000 isn't tree3, that's the number of symbols it would take to write down a perfect proof that tree3 is finite
7:40 I'm surprised the paper didn't implode into a black hole destroying the entire universe from what you just wrote on it 😂😂
I love how excited these guys get about this stuff!! Very interesting
4:22 "the universe will eventually reset itself" "reset itself"
This is one of the few TH-cam videos that I watch over and over again. I'm iterated.
I love how excited he is! You can see he just loves math
4:23 There was a glitch in the matrix.
John Thimakis It happens when they change something.....
That was the universe resetting itself
Wait a glitch in the matrix? glitch in the matrix?
Was it the same gesture or different gesture?
If the universe did reset itself, how would we know?
When he started trying to top TREE(3), I almost had a panic attack.
I love how the extra footage is longer than the original video
This guy's enthusiasm is contagious!
What's the most number of nodes in any tree in TREE(3)?
Sarthak Bansal TREE(3) means three types of nodes. Not nodes in general.
Sarthak no it's 3 colors of nodes, the nth tree can have n nodes, but they can only contain 3 colors.
It would be 1. As with TREE(1) and TREE(2) you only use one of the single seed options until the very end. Once you have no options that don't include a previous tree, then you use your single seed options. If you use them at any point before the last two, then they will appear in other trees immediately, thereby ending the game prematurely.
Octagonalsquare that was not the question though, his question was as followed.
What is the global maximum f(x) on the curve that is the curve of nodes pertaining to each iteration of x in the well defined function TREE(n) when n does equal 3.
Now as far as I'm concerned the upper bound to that question is
TREE(3)^(1/3)
That is the last tree Octagonalsquare, not the largest tree.
I literally hear GNASHING OF BOLTS HOLDING EDGES OF THE UNIVERSE when he started making TREE of TREEs
Mathematics really feel like magic. By playing a simple game on a piece of paper, you can actually write a concept that is bigger than existence itself. This is mindblowingly elegant.
Bravo on the cliffhanger from the first video to the second
And yet it's true to say, almost every integer contains tree(3)
tdurran That's something to think about.
And almost every positive integer is bigger than tree(3)
Woah yeah ... wow
a stupidly mindblowingly correct statement
Darn you for exploding my head
If you take tree(3) and substract 10% of it, and add all the numbers together, and then add all the numbers together, and so on as long as it will be just one number I bet this number is 9. 😊
Man, I love this guy! Big up Tony!
Sometimes I get the unexplainable nervousness when they write down things like TREE(TREE(3)) as if the universe is gonna explode simply by some sentient being conceiving those numbers.
i think the awesome part of Tree(3) and some other large numbers is that they were not discovered with the intention of finding a large number. im not a part of it but in the Googology fandom there's all these efforts to create simple mathematical situations that give large numbers, but i just like to imagine that, when studying these trees, someone just accidentally stumbled upon Tree(3). its not even close to being as large as Tree(3) but the Monster Group is one of these; a fundamental building block of groups with just completely unexpected size and connection to modular forms
"...how quasi is your ordering?"
.."It's well quasi mate"
His neck tendon pops out while he talks. These guys are so beautifully passionate.
The universe reset itself: an overflow error probably occured.
I'm going to say it... be prepared... because:
We gave birth to the tree function, we chopped it down to the log function, and it was so naturally done (ln) that 'i' celebrated it with ∑ π.
😂😂
The universe must of reset 3 times before you thought of that
Jorge C. M. I prepared to be mind blown...
But I wasn’t...
Because I have no idea what that means.
Omg I love math jokes.
What's sigma? I understand it means "a portion of" but what's the textbook word for that function?
I got a bit lost after "tree"
Dude, thank you for making maths fun to listen to!
Tree (tree (3)) makes my heart heavy
"Universe resets before you can complete the proof" Awww....There goes my plans for the weekend..
These numbers just embarrass the size of space-time.
I love the quick reset of "The universe resets itself." Well played!
I just wanted to find out how big TREE(3) is, not have an actual existential crisis about the universe resetting itself.
I love this guy, please make an interview about his life interests... PLEASE XD
Subscribe to the Numberphile channel and you'll know...
Raf M. I am, and know tidbits from him, but I don't know... Where does he get all this interesting topics if he works on physics. How does he know so much math, or is it not much, just what is asked for theoretical physics?
Isn't he a Liverpool fan?
TREE(Graham's number) did I break the internet?
...TREE(TREE(TREE(Grahams number)))... Until TREE(Grahams number) amount of TREE functions put into ...TREE(TREE(TREE(Grahams number)))... amount of TREE functions ...TREE(TREE(TREE(Grahams number)))... times
Stumbling across these numberphile videos in 9th grade, I for once was curious about something related to math. "Related to math." I didn't realize at the time that this *is* math, and this is largely how math feels to mathematicians. Exploratory, creative, boundless, surreal, and objective??? All at once? Wow. Fast forward a few years, and I'm just obsessed with math. I'm a math major. Thanks for the awesome videos!
What if you could prove tree(3) was prime...
Call them when you have supersublinear algorithm.
2^(13*TREE(3))-1=Big number, could be a prime. Maybe not.
@@ydg7670 100% not. There's a proof it's not.
What the heck? Got the notification for this before the actual video!
Don't you hate it when the universe resets itself at the wrong time.
Yes, this is how I explain that some things truly are impossible: when there isn't enough energy in the observable universe to do the thing, even if you used it all, with no losses, and did it as perfectly as it could be.
Still remember the time when I first learn about a number called Trillion and that blown my mind and here are we now.
My mind is not abstract enough for this. I kind of get it when he explains it but I’m like “but how do they *know*?
When he says the universe resets itself, the running frame in the video resets itself. Funny trick! :D
I love this guy
I love the sense of megalomania that unimaginably huge numbers inspires in Tony.
In all the gee whiziness about the size of the forest Dr. Padilla neglected to mention the, to me, fascinating fact that the tree(3) forest contains only one green node.
Steve's Mathy Stuff well, it is not necessarily green, it could be black, or red, or maybe blue, or even purple
I think we can just get away with assuming, without loss of generality, that it is green.