Goliath & Leviathan Numbers - Numberphile

Featuring Tony Padilla. See all three videos in this Apocalyptic Trilogy - bit.ly/ApocalypticTrilogy
More links & stuff in full description below ↓↓↓
The other two videos in this trilogy are:
Apocalyptic Numbers - • Apocalyptic Numbers - ...
Primes and Fibonacci with 666 digits - • Interesting 666-digit ...
Extra James Stirling snippet - • James Stirling (extra ...
Tony Padilla on Amazon - amzn.to/3U6DRSM
(Check out “Fantastic Numbers and Where to Find Them”)
See our 666 Playlist - • 666 on Numberphile
Bibledex - / bibledex
Big Numbers Playlist - • Big Numbers on Numberp...
Patreon: / numberphile
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Our thanks also to the Simons Foundation: www.simonsfoundation.org
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The picture of demon legions were AI generated and are not real demons.

[@numberphile]

All three videos in this little trilogy at bit.ly/ApocalypticTrilogy

[@JohnLeePettimoreIII]

tree goliath

[@WhataMensch]

Hey we love you America Numberphile and I want to say the UK is in my thoughts as you deal with the loss of the 3 aid workers accidently hit. Love you guys

[@IDFpartyboi972]

@@WhataMensch I sat on the joystick by accident. sorry.

[@twixerclawford]

Absolutely unbelievable that he didn't mention the leviathan (rounded) would literally be 666E666

[@phoquenahol7245]

The log base 10 of that but point stands.

[@Christian.L.Rodgers]

For those interested, the name legion 'Legion's Number' is a reference to Mark 5:9. Jesus asks a demon it's name and it replies "my name is Legion, for we are many."

[@nickfifteen]

I thought that was the nickname of the notorious hacker known as "4chan"? /jk

[@sethv5273]

That’s a cold response ngl

[@colinpountney333]

Well, you would not expect a demon to be called Nigel.

[@17.11Acts]

I wonder what Ivan Panin found on this? Hummm

[@piewithinapie]

The log_10 of the leviathan number seems suspiciously close to 666*10^666...

[@dskinner6263]

Tony Padilla's version of a super leviathan number being smaller than Graham's number, and that being smaller than TREE(3), does give one pause.

[@JohnSmith-nx7zj]

Factorials only have roughly the same strength as exponentials. So a power tower (with some factorials thrown in) of factorial height is only about a triple up arrow in strength. I’d image G1 is bigger than Tony’s leviathan number.

[@friiq0]

John Rickert was once a professor of mine! He is such a sharp guy and a wonderful teacher. I remember that he held his students to a high standard, and his enthusiasm for math was very infectious to me.

[@brouquier7172]

Since Tony mentioned the Stirling approximation, this video ought to give an inspiration for another series of videos: interesting approximations in the world of mathematics. I know some approximations were covered in previous numberphile videos, but why not make a series out of them? Brady please do your magic!

[@nicholascmcneill]

I love Tony Padilla and his big numbers. One of my favorite bits of numberphile

[@docsigma]

Mathematicians have officially “gotten ridiculous” (and I for one am all for it)

[@smylesg]

5:29 I think for this audience, it's a little "horned" thing.

[@ivanhadevych]

9:44 "The horny number" 😂

[@Matthew-bu7fg]

Imagine how powerful David's sling would have had to be to defeat Graham

[@Filipnalepa]

No matter how big N is, if hit with with *0, it falls.

[@nidoking042]

Interesting to see Dr. Rickert in this video. I studied under him, very briefly - never took a full class, but he went over answers to a Putnam competition and I watched him do math way over my head for about an hour.

[@sthelenskungfu]

I miss BibleDex. I get that I'm probably the only one, but it's great to see Bible scholars interviewed without the kind of hype that other places give.

[@17.11Acts]

Ivn Panin! Have you heard of him?

[@sthelenskungfu]

@@17.11Acts No. Should I?

[@HenryCrun2]

I assume the Legion connection is from the "we are legion" quote from a demon in the bible?

[@stigekalder]

Yes another commenter says it refers to Mark 5:9.

[@demonologus]

That was awesome, i love these very huge numbers!

[@numberphile]

Glad you enjoyed it

[@ViliamF.]

When will this video be un-unlisted?

[@Bunnokazooie]

Loved Bibledex years ago!

[@glowingfish]

I wonder what this channel is going to do on April 20

[@unvergebeneid]

1:44 have we run so out of symbols that we are using Sindarin letters now?!

[@AlistairBuxton]

I think it's ᚠ (Fehu) from Futhark (Norse runes).

[@wbfaulk]

Especially since ᚺ and ᛞ are also Nordic runes.

[@tomkerruish2982]

Cirth does certainly resemble fuþorc.

[@unvergebeneid]

@@wbfaulk makes sense then.... _sad Gandalf noises_

[@michaelpenklis3104]

If it's called a Goliath number, no matter how big it is, you can break it down with the right knowledge

[@romanvolotov]

this is so silly and goofy, i love it

[@pennnel]

Since it appears in the last 3000 digits, and they don't know if it's the smallest. I'm guessing that they discovered it by only keeping the last ~3000 digits for each multiplication of 2, and they just kept going until the goliath sequence appeared? So it's proof of a goliath number, but they only checked the last ~3000 digits of each value of 2^n. Probably used a higher digit count than 3000 to check, but it illustrates my point.

[@therealax6]

You can't check that many powers. That exponent is big, as in "bigger than the number of particles in the universe" big. There's no way to check them one by one - there has to be some way to generate possible candidates.

[@PabloMoleman]

How many digits does the smallest known Goliath number have in total? Tried to Google it but could not find it, or any other sources on it, for that matter. Any more info?

[@coyets]

It was stated that Graham's Number is larger than the numbers mentioned in this video, but Graham's Number is notated as G64 because it is the sixty-fourth number in the defining sequence. Would there be any apocalyptic significance of the number G666 in that defining sequence? Would there be any apocalyptic significance in using the number 666 rather than three in defining the sequence analogous to the sequence used in defining Graham's Number?

[@kuenmao]

7:20 Since we're just looking for a comparison, there's a much easier approach that doesn't require Sterling's approximation. By considering the expanded form, one can see (1E666)! is significantly less than (1E666)^(1E666). Then we can take the logarithm: log((1E666)^(1E666)) = 1E666 * log (1E666) = 1E666 * 666, which is clearly less than the other log value, ~1.6E1596 (at 7:02).

[@thirstyCactus]

I get goofing around with devil numbers, but what use is super factorial?

[@picrust314]

Is there a paper on finding goliath numbers that is accessible?

[@malicorx]

have you guys ever tried to find the "big"-number that is closest to Grahams Number ? (and i don't mean Grahams Number -1, but some number that has a different way of construction)

[@Ms.Pronounced_Name]

For the purposes of definition, would 6666 be 1 or 2 666's?

[@davecorry7723]

Brady's such a watch-dude. Think I saw him with a lovely Sub before, and here he has a moonswatch. 👏

[@jamalabdo4660]

The way that phone charger is folded next to the laptop is so uncomfortable

[@carloselfrancos7205]

New numbers dropped 🔥🔥🔥

[@trifonmag4205]

Imagine if you subtract 1 from the smallest known Goliath number, you get a mersenne prime lol

[@LeonidGeorgievIND]

INDETERMINACY OF All-Power Measurement. Correct me if Im wrong 😊

[@grafzhl]

Might it be worth to use a less shallow focus when brown paper is involved? My impression is that in recent videos there's a lot of focus pull that, to me, is very distracting.

[@lol-hy4mk]

Can't believe Jim Sterling was a mathematician before making videos about videogames

[@LuxiBelle]

His life really is in perpetual decline

[@jb76489]

Don’t compare a mathematician to that loser

[@mrosskne]

?

[@lol-hy4mk]

You know, the jimquisition guy. He dresses like a clown and "is" pregnant.

[@TimothyReeves]

Don't know about the video game guy, but the mathematician is Stirling, not Sterling.

[@tomasarana8450]

"Devil sized region of space" would be a great album name

[@tlou34]

I love this man

[@robinbrowne5419]

Are there an infinite number of weird numbers and sequences?

[@Erebusdidnothingwrongish]

I asked Siri and checked Google and Yahoo this question could not be answered. I cannot find a post on Reddit or on Google..?

[@peterflom6878]

We need David numbers and slingshot numbers!

[@numberphile]

Nice.

[@Sairin13]

Very much like the actor who played Wormtail in Harry Potter, or the guy minding the Red Dwarf Back to Reality game

[@CC-gg4oj]

It's nice to know even Tony crashes computers trying to compute numbers. It's not just me! He even crashed his paper...

[@samtux762]

How to compare large numbers? Power tower does not help.

[@ggs4989]

It bothers me when people say stuff like that about doppelgangers. Assuming there were infinite universes, there's no reason to assume that you would have all possible configurations of matter. You could simply have an infinite number of universes with a single hydrogen atom, or any other arbitrary configuration.

[@ianmoore5502]

5:30 the most wholesome laughter

[@froggy_days]

This is a bit weird big number to focus on, right? You could also focus on big number theory in general

[@kasperlindvig3215]

Why are you using 6's ?

[@valeryrubantsev7837]

The crow can only count to 5, so they are happy.

[@Hadar1991]

I am kinda missing the point. Those numbers seem to be completely arbitrary (in mathematical sense). To they have any proper mathematical properties or this is just some "funny exercise", you could repeat with other "popular" numbers like 69, 420 or 2137. :v

[@rjtimmerman2861]

Exactly what they are, and I don't really care about them

[@SauloMansur]

It's a funny exercise, but also serve as a sandbox for the development of mathematical methods. For example, how do you find a sequence 666 in a power of two? Then you go on trying to break this problem into smaller steps, find shortcuts in calculations, use approximations for lower and upper bounds, and so on. The funny exercise may not have a purpose by itself, but give a well defined problem to work on.

[@SgtSupaman]

He mentions that in the first video of this trilogy. This is all just for fun. It can be generalized to other sequences of digits, but he doesn't care about any potential future application (or lack thereof).

[@skasperl]

Find someone who likes you as much as Tony likes large numbers.

[@seventhtenth]

where i can i learm more

[@KiloOscarZulu]

Why aren't there any Behemoth numbers? If you are talking Biblical monsters and mention Leviathan, surely Behemoth should get a mention.

[@jimbenge9649]

I'm just fascinated by big numbers. Shame my bank account isn't. 🙃

[@Killer_Kovacs]

The Goliath numbers symbol looks like a Norse rune

[@wbfaulk]

Since the next two also look like Norse runes, I'd expect that was the intention.

[@pacman52280]

Instead of trying to find 666 instances of 666 consecutively within the fully written out power of 2, I wonder what the smallest power of 2 is where they DONT have to be consecutive.

[@landsgevaer]

Hmm, on average a 6660 digit long number should have about 666 6s. Then you are close to 2^22k. You could make a better estimate about how much sooner it could occur since there are lots of lower powers that could by accident have an overrepresentation of 2s. But I bet that it isn't far off the 22k-th power of two. A computer can do that brute force for you...

[@landsgevaer]

I checked it. 2^20674 has 666 6s in it. A bit smaller than I expected.

[@pacman52280]

@@landsgevaer, read my comment again. I didn't say 666 6's. I said 666 666's, non-consecutively.

[@landsgevaer]

@@pacman52280 Ah, but then you can surely extrapolated my estimate to this problem too? I'll leave it as an exercise for the reader that that occurs around 2^2M. 😉 Takes a bit more effort for a computer to do, but still possible...

[@guru0503p]

Camera work was all over the place towards the end of the video

[@garygenerous8982]

I’m kinda surprised nobody commented on (6^6)^6 as an apocalyptic number.

[@macronencer]

When people talk about the inevitability of duplicates in a sufficiently large universe (or similarly, the idea that every possible digit sequence might appear in pi), I'm never satisfied with the argument. I understand that a sufficiently large universe will have SOME duplicates... but then the next claim after that is that EVERYTHING has a duplicate. What's the missing step? I remain unconvinced. What's the problem here? Is it that there IS a rigorous proof but it's too hard to explain in informal ways so nobody's managed to communicate it yet?

[@friiq0]

It’s a statistical argument. Think of rolling five dice. There’s only a finite number of possible outcomes, right? Well if you roll the dice enough times, you would expect to get every possible combination eventually. And if you kept rolling, everything would be a repeat of something that came before since every combination has already shown up. If you just keep rolling over and over again, it would be weird if certain combinations repeated but others magically never showed up again. In the case of five dice, there aren’t that many combinations, so it wouldn’t really take that long in the grand scheme of things to start seeing repeats again. If instead of rolling 5 dice you start talking about ways to arrange atoms in a finite region of space, note that there is still a finite number of possible combinations. If you have a big region of space, there could be a truly vast number of ways to arrange atoms in that space, but it is still finite. So, if you randomly arrange atoms in that space over and over again an even more extremely vast number of times, eventually you will hit every combination. After that point everything will have to be a repeat of something before, and if you run through that cycle multiple times, it would be weird if certain things repeated but others seemed to magically get skipped. That wouldn’t be very random. Granted, the universe itself isn’t completely random. There are big clumps of matter in stars and then vast empty spaces between them. However, there is some randomness in the universe, and that provides a bit of wiggle room in these arguments about duplicates in sufficiently large universes.

[@macronencer]

@@friiq0 Wow! Many thanks for taking the time to give a detailed answer, I appreciate it. I do understand what you mean. I think that when people talk about this they often forget to stress that it's a statistical argument, entirely predicated on a random process to generate the combinations of stuff. In the case of physics I suppose that's reasonable, although when it comes to things like every finite digit sequence appearing in the decimal expansion of some transcendental number, it gets much tougher because mathematical assertions require rigour.

[@friiq0]

@@macronencer You’re welcome! I love this kind of stuff. I see what you mean about the mathematical arguments. I think that mathematicians have proved that pi and other transcendental numbers will never settle into any kind of regular pattern. Even though the digits are fixed, that’s why you can think of the digits of pi as “random”. The argument is also different in the case of pi, because the digits go on literally forever. Suppose we consider pi in binary. We expect that pi will eventually write out the entire play Hamlet by Shakespeare at some point. Why are we so sure? Well, suppose that Hamlet never ever shows up in pi even once. That would mean that pi goes on for an actual eternity and somehow magically dodges Hamlet for ever and ever and ever-even though Hamlet is just a finite string of digits like any other. This is why mathematicians are quite confident that pi will eventually contain Hamlet. They are confident there is no special ordered pattern to the digits, and so there is no special Hamlet-avoiding pattern in the digits either. The same argument works for any play, book, poem, etc.

[@macronencer]

@@friiq0 I know what you mean. Is "confident" the same as "certain", though? We're getting into the gritty areas of metamathematics if we start down that road, perhaps...

[@friiq0]

@@macronencer I don’t know the math well enough to say that we know for absolute certain. Maybe someone has proved it, but I couldn’t tell you specifically.

[@LIA-52]

11:22 David would need a way bigger slingshot to beat Graham....

[@davidmilhouscarter8198]

7:38 Why does “e” show up in so many equations?

[@primeedits840]

How big os no. 'Tree'?

[@helgefan8994]

I believe I found a mistake: At 7:00, the number 1.6 times 10^1596 is certainly *not* bigger than Googol-plex, which is 10^(10^100).

[@nickfifteen]

Yeah that one threw me for loop as well. It's definitely larger than a googol (10^100), but definitely not a googolplex (10^[10^100])

[@pauldubois0]

But it's bigger than log gplex, which was the point

[@helgefan8994]

@@pauldubois0 Dang you're right. I get it now, thanks!

[@brendanmitchell4581]

Nobody is talking about how the symbol for Goliath numbers is the letter G in Tolkien's Cirth alphabet! Mathematicians are getting creative with their symbols :)

[@wyattstevens8574]

I recognize that symbol for Goliath numbers- it's Gandalf's signature from Lord Of the Rings!

[@brendanmitchell4581]

It's the letter G in the Cirth alphabet. Mathematicians be getting bored :)

[@joshuaevans4301]

I laughed out loud at the definition of a super factorial. That's just ridiculous

[@stapler942]

A Goliath of Leviathans is a Golviathan.

[@judychurley6623]

"My name is legion" appears somewhere in the bible. (Lucifer is speaking, or one of his minions, as I recall.)

[@jameshiggins-thomas9617]

Saying that's the smallest known Goliath # implies there's a known larger one? How many are known?

[@MrBeezweeky]

David will find the smallest Goliath number

[@user-wq9bb4jl7t]

The word Goliath جالوت appears in the Qur’an in 3 verses Total sum of verse number from the end of the chapter = 111 111 * 6 = 666 and The gematria of these verses = 30522 30522 = 6 * 666th prime (4973) + 666 + 6 + 6 + 6

[@robertdarcy6210]

Any resistance PS3 fans here? Remember the Goliaths and the Leviathan? ❤

[@TheSqueak788]

Why did Tony not round the Sterling approximation to 666x10^666?!?!

[@yasedb]

This ia actualy hilarious

[@WhataMensch]

Huge fan of numberphile from America and I want to say I am so sorry for your 3 aid workers. Love from America in your time of grief.

[@IDFpartyboi972]

Sorry still had sleep in my eyes from just coming on shift.

[@salmiakki5638]

Don't be sorry, it's the 3*10⁹$ you just gave them and the NATO arms stockpiles in the region that allowed it to happen. Biden could stop making angry remarks tomorrow if he only started heading back the military support

[@hybridvigor4594]

@@IDFpartyboi972The angry mob is at your gates, Dr. Frankenstein.

[@NStripleseven]

Funny thing about the Leviathan number is that you can write it as 10000000000000000…00000000000000000.

[@blumousey]

I suppose this is all arbitrary, because we are choosing base 10 to do these calculations. Maths is dispassionate towards what base to use.

[@JGMeador444]

I had that thought through the video as well, and I'm surprised it wasn't mentioned.

[@Nebukanezzer]

Long sequences of repeated digits in a given base are interesting, however you choose to find them can be extended to other bases. Plus, you didn't seem to think about the choice of powers of 2 also being arbitrary.

[@JGMeador444]

@@Nebukanezzer You're right, I didn't question the exponent's base being 2 because it was stated in the definition of the number. Yes, it's arbitrary, but it's clearly defined. Also, in the video they mentioned that you could consider other bases for the exponent and do the same thing. On the other hand, the "in base 10" part was never explicitly stated and was simply assumed throughout the video; that's the part I have a problem with.

[@SauloMansur]

​@@JGMeador444The choice of 666 was already arbitrary, and the base 10 came from this. The number and base are just a motivation for the problem, making them funny to work with, but the mathematical tools applied are not limited by them. You could ask "what is the smallest power of 3 where a sequence 3457 (base 8) appears", and the methods would still be the same. But nobody cares about 3457 in base 8, so there would be no fun doing this xD

[@terranceparsons5185]

Has anyone watched QI when Stephen Fry claimed the number of the beast is actually 616?

[@MagruderSpoots]

These big numbers make me realize that the universe is actually small.

[@inquaanate2393]

I don’t think this maths really has a purpose 🤨

[@l3zl13]

10:42 sums up the whole video

[@nightwishlover8913]

But WHY?

[@GrimAce6615]

So leviathan number is ~666*10^666 🤔

[@user-wq9bb4jl7t]

The number of Goliath contains 1998 consecutive numbers of 6 just before it we see the digits 4214 What is strange: In the Qur’an: Verse No. 4214, its gematria value = 1998 -------- The word Goliath جالوت appears in the Qur’an in 3 verses Total sum of verse number from the end of the chapter = 111 111 * 6 = 666 and The gematria of these verses = 30522 30522 = 6 * 666th prime (4973) + 666 + 6 + 6 + 6 ------- The last appearance of the word Goliath “جالوت” in the Qur’an was in chapter after 1075 words from end The strange thing is that verse number 1075 in the Qur’an its gematria = 666 ---------

[@DOTvCROSS]

Wow I found it. What? The number that I said to be significant only to let others know I found it.

[@jacksonstarky8288]

Legion's Number should be 666(pentation)666... and I'm not satisfied with the conception of Leviathan numbers at all as defined, simply because of the use of 10 as a base. But I'm with James Grime, and very much anti-decimal, for many of the same reasons.

[@MrSaywutnow]

I have to ask - is there such a thing as a Chad number?

[@robsquared2]

"I went to the place where it happened." Well, no, no you didn't.

[@LadymViter]

We need a Cthulhu number

[@seancooper4058]

I'm glad Numberphile finally covered the horny number... finally putting the "phile" in Numberphile

[@nickfifteen]

YOSEMITE SAM: "Great horny numbers!"

[@neotericrecreant]

How many other people got an ad for some sort of exercise equipment before this?

[@fs-DRAITHEO]

Numbers required a Decimal point to represents the Tenth. So are them Numbers Whole or Fractions? F. Symbol used represents the Bass Clef on music sheets.666 is FFF Dymanic markings meaning VERY LOUD.

[@DeathSugar]

> trilogy > 4 videos lol

[@graduator14]

5:29 That's what my ex called me!

[@deadmanrang]

Why is this unlisted?

[@unvergebeneid]

Sequels always start unlisted and will be de-unlisted after a while.

[@deadmanrang]

@@unvergebeneid fair enough

[@JavedAlam24]

@@deadmanrang Maybe for statistics? See how many people click the links?

[@Tahgtahv]

My assumption is that since this is a series, it makes sense for public recommendations to not include later parts.

[@satyayuga0]

​@@Tahgtahvit's called a 'bug' in the program

[@asdfoifhvjbkaos]

gigachad in thumbnail

[@One-mx2rk]

Hey, those images for the legion numbers seem to have been created with generative AI. Is that so?

[@deadmanrang]

Now that Brady has mentioned it. I'm still sad that Brady hasn't continued the Bibledex series!

[@JxH]

Focus is becoming an issue with these videos. No, I don't know a solution...