Six Sequences - Numberphile

Which of these number sequences do you like best? Vote at bit.ly/Integest...
The extra bit of footage is at: • Tony's Favourite Numbe...
More links & stuff in full description below ↓↓↓
This video features Tony Padilla from the University of Nottingham: / drtonypadilla
Here's each sequence on the OEIS:
Khintchine's constant: oeis.org/A002210
Wieferich primes: oeis.org/A001220
Golomb's sequence: oeis.org/A001462
Largest metadrome in base n: oeis.org/A023811
All 7's: oeis.org/A010727
Wild Numbers: oeis.org/A058883
The Aperiodical: aperiodical.com/
Brown Papers: bit.ly/brownpapers
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[@Majorohminus]

my favorite sequence is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10... its the natural sequence and its perfect. the number in the nth position is n and its the first sequence anyone learns.

[@todabsolute]

@Adi Septiana 1. It was supposed to be sarcasm 2. This sequence is the base for e

[@aforcemorepowerful]

It's also the decimal expansion of Champernowne's constant

[@TheMagicianLiam]

I agree but in base 12. Sorry

[@hkayakh]

How about that sequence but nth position is -n?

[@sankang9425]

Ah yes, the A000027... My second favorite.

[@m3ntalcas3]

i could tell khinchin's constant was his fave he went on about it much more than the others

[@leo17921]

also cause its more complicated

[@rednecktash]

thats what i thought too even before seeing any other ones

[@thomaskaldahl196]

But why the decimal expansion? Is there anything special about it?

[@olivialuv1]

@@thomaskaldahl196 The decimal is cool bc you get to know the approximate value of this godly self-knowing number, as opposed to just some fraction whose value you can't tell by looking at it

[@thomaskaldahl196]

@@olivialuv1 But what's significant about base 10 as opposed to binary or some other base?

[@christosvoskresye]

It would seem to me that the constants in the continued fraction expansion of Khinchin's constant would be more meaningful than the decimal expansion.

[@RedRad1990]

*Tony Padilla:* "I'm not going to tell you which one I like best" *also Tony Padilla:* proceeds and starts by talking about his fav no hints XD

[@numberphile]

Shout-out to Ireland!

[@goutamboppana961]

??????????

[@overwrite_oversweet]

This virol ad said something like 95% of KZbin vid get less than 1000 views and my first reaction was "wow so many KZbin vids get over 1000 views"

[@anticorncob6]

One of my favorite number sequences is this: 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, etc. It's all the primes, plus all the values of p^(2^n) where p is prime and n is a positive integer. With these numbers, every other number can be written as a unique product of these, without repeats. For instance 99 = 11 * 9, that's the "factorization" and there's no other way to do it.

[@DimaVaulin]

Wow!

[@bagelnine9]

Same.

[@numberphile]

In the video description there are links to all the sequences, a chance to vote for a winner and other stuff...

[@mr.z111]

Ш vs Щ

[@gmdFrame]

@@mr.z111 Прив

[@numberphile]

Do you know what - I kind of get it and never really mind it.... It is human nature to get a thrill from being first (or among the first) to do or see something...

[@WilliametcCook]

2:53 But I thought God's Number was 20... James Grime was in your video on it...

[@annevanderbijl3510]

Nice

[@ToxicGLaDOS]

Could you do a video on the Tree function? I've looked up some things about it but it's over my head without intense explanation. Mostly about TREE(3) and how it compares to grahams number and other big numbers.

[@alecbader7433]

You were ahead of your time...

[@asheep7797]

4 years ahead of your time.

[@unecomedy13]

for the wild numbers, just add 0.5.

[@NeoUno866]

1,11,21,1211,111221,312211, ... You split it up and describe the previous number, where the next number in the sequence is the description.

[@peligrosacurva-cz4ev]

See and write 🎉

[@smoorej]

Khinchin’s constant is absolutely mind blowing. That any continued fraction expansion of “almost all” numbers gives you Khinchjn’s constant is just jaw-dropping. Question: is the “almost all” numbers all real numbers except the rationals?

[@vocnus]

It is really incredible and yet ture for ''almost all'' numbers... however it is NOT containing each and every irrational number! For example fi=1.618... or the base of natural exponential e=2.718... are irrational numbers which are not under this rule. The fi's fractional expansion goes this way: [1;1,1,1,...] which is the notation for 1+1/(1+1/(1+1/(1+...))), and with the ''e'' it goes this way: [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] = 2+1/(1+1/(2+1/(1+1/(1+1/(4+...))))). In the first example the geometric mean is constant 1, and in the second it goes to infinity as we deal with more and more terms...

[@Catman_321]

you can show pretty easily that no quadratic irrational number has this property. Since the terms in the continued fraction repeat periodically, it will not converge to an irrational number. Same with numbers where the terms in their cf strictly increase, which diverge to infinity. e is similar to these numbers and doesn't converge to khinchin's constant for similar reasons however most generic irrational numbers do have this property

[@GladionD.Pierce]

NERD

[@joegamewaffle5967]

False, the best name is not Solomon Golomb. It's Nebuchadnezzar, solely because it's pronounced something like "Ne-buh-kuh-ne-zur." ...My Global Studies teacher is quite awesome...

[@jikojj3680]

Joe GameWaffle global studies 🤣 ought to be usa uni

[@amoledzeppelin]

Yep because most of other world learn how it's pronounced from watching the first Matrix movie in original.

[@maxnullifidian]

The best name might well be Arup Gupta - at least according to Click and clack, the Tappet brothers...from Car Talk. ;-)

[@benjaminnewlon7865]

Hes in the bible

[@CraftQueenJr]

Walt F. Those end credits were always hillarious.

[@AymanB]

Solomon Golomb ! That is a great name.

[@numberphile]

hello!

[@jadenchen8921]

Hi numberphile!!! I’m a huge fan

[@annevanderbijl3510]

@@jadenchen8921 hi

[@happypiano4810]

Hi!!!

[@googelman]

ae

[@ajeydevadiga6652]

numbers are just awesome......what a beauty...

[@ChristianPerfect]

Thanks for your votes, everyone! Golomb's sequence won the vote, but the only sequence we could fit on the trophy was the Wieferich primes so we said that won instead. Look at the trophy on The Aperiodical, it's magnificent.

[@whatno5090]

My favorite integer sequence is and will forever be the look and say sequence.

[@IMortage]

Numberphile had Conway himself talking about the look and say sequence

[@ckmym]

1 11 21 1211 111221 312211 13112221 1113213211 ...

[@Cernoise]

Maybe I misunderstood something, but if the first one is just the sequence of digits in a real number, and the integers in the sequence aren't actually used as numbers, it's not really significant as an integer sequence. It's not s very integ sequence at all, let alone the integest.

[@Ensivion]

I love how Tony's collar was popped for most of this.

[@VeteranVandal]

I knew it was the first one, because I knew he would just be able of holding himself in the sequence he liked the most if it was presented first.

[@Leadvest]

Could you do a video on A027746? It's a list of n by prime factors.

[@kurzackd]

7:20 -- umm... in unary numerical system, 0 is actually ... empty. Not 0. Just ... nothing. No numbers. Empty string.

[@Onoma314]

I'd love to see these guys do a video on the 216 digit Shemhamphorasch. It's produced using only the 24 digit reduced Fibonacci series and the numerals 1 - 9

[@Epaminaidos]

The one I like most: 1, 11, 21, 1211, 111221, ...

[@XenophonSoulis]

312211 RIP John Conway

[@evanknowles4780]

13112221

[@manmanman4825]

@@evanknowles4780 1113213211

[@Lahbreca]

Why was 67 twice in the wild numbers, if that was just the list of numbers that would result from the operation being done on any given number?

[@doublelxp]

I was thinking a more interesting "all the sevens" would be 7 in each of the bases, but it would just be 111, 21, 13, 12, 11, 10, 7, 7, 7, etc.

[@Spitzenhund]

Its a nice technique that helps with concentration. We are trained to see brown as a constructive material so writing on it makes us think we are doing more than just writing on paper. The tactile sound and feel of the paper also helps with concentration and I honestly think it sounds nice and prefer it over just normal paper or a white board

[@CoyMcBob]

Couldn't you start Golomb's sequence with a 0? 1 appears 0 times. it would be: 0, 2, 2, 3, 3, etc.

[@earfolds]

0 would appear one time, so the sequence would be 0, 1, 2, 2, 3, 3...

[@CoyMcBob]

No, there is no zeroth place, and each element tells you about the next element, not the previous. The first element will tell you that there are 0 ones. Thus you know that the second element has to be a two, and so on and so forth.

[@CoyMcBob]

In fact, even more than that, you can just dump zeros wherever you want. Examples: 00000666666777777... is completely valid. So is even 10005555566666 and so on. This sequence only works out if you do it by indexing and not by counting, or you define the range as n >= 1

[@earfolds]

I guess that would work, but then you could even argue the sequence is valid as a string of infinite zeroes. The most interesting one is the one with positive nonzero integers only because zero is trivial.

[@CoyMcBob]

Or, you could just define the sequence as one that every single number in it is referenced to in the sequence. That would also work.

[@TheIcyBlade99]

yeah when I see it I think its annoying, but when I am first comment I do feel the temptation to say first haha

[@ZipplyZane]

I had the same problem, and I wasn't using the subtitles. It just sounded like descending to me. I think it's that T at the end of strict being right by the A at the start ascending. It sounds like strict-d-ascending.

[@fearingfearitself]

You didn't unpack the formula behind all the 7s: 7 x 1^n, where n in the position in the sequence.

[@annikapeterson4061]

Can there be a whole video on Fermat's Last Theorem?

[@PC_Simo]

9:11 There *_IS_* an equation to that: a(n) = 7; and it rhymes, too 😊.

[@Devilogic]

When talking about real numbers, "almost all" is typically defined as "all except for a set of (Lebesgue) measure zero". This is the case here as well. The exceptional set here is in fact uncountable! A subset of it is the uncountable set of all reals with only 1 and 2 in their cont. fraction expansions - the geometric mean will be less than (or =) 2, but Khinchin's constant is >2 Another is the uncountable set of reals with numbers >=3 in their expansions - the geo. mean will be >=3, but K0

[@peligrosacurva-cz4ev]

8:17 how to get 194 in this sequence? 5"3+3*5+5"2+2*5+5"1+5+4=189. No 194

[@peligrosacurva-cz4ev]

I got, I got 125*1+25*2+5*3+1*4=194 😂😂🎉🎉

[@AMotoVlogger]

@Numberphile look out for a paper with a conjecture on Pi and the "All the Seven's" coming to a computer near you.

[@sdvalen7761]

Levy's constant applies to itself and "almost all" numbers in the same sense that Khinchine's does. It's a related property of continued fractions. I nominate A087602 (its decimal expansion) and A086703 (its continued fraction expansion) as my favorites.

[@thoughtyness]

What is the proper notation for the continued fraction at 1:10? If inputting into WolframAlpha, etc how would you correctly write it?

[@MinusPi-p9c]

Probably just explicitly. a0+1/(a1+1/(a2+1/(...)))

[@lordofedges6487]

Meanwhile at computerphile... "Ok, we have one sequence, it goes like this: 2, 4, 9..."

[@electroflame6188]

More like 1, 6, 21, 107, ...

[@keyofamajor]

aha, thanks! looking up "almost all" on wikipedia says that there are "a number of specialised uses" of the term, which continues to confuse ._. definitely not as bad as "mathematical concepts named after leonhard euler" though

[@jamespfp]

I called his favorite after he described its self-referential completeness. Ascribing divinity to it -- I tend to think of that as sentimentality, but it also gave me a chuckle. Nothing is as charming (at present) as completeness, eh?

[@kurtiswithak]

What about this sequence? 1, 11, 21, 1211, 111221, 312211, 13112221... It starts with 1 and the next element "describes" the previous, hence the second number in the sequence is "11" meaning "one 1" (describing the previous number). The third number is then "21" meaning "two 1s", and so on. I'm sure there's some very interesting math regarding this sequence..

[@cecasiahaan6801]

Kurtis Fraser 13112221,413213,21122314,31321314,31123314,13123314,13123314,13123314,loop.

[@MrProfetZ]

Conway did some work on this sequence, I think numberphile even has a video about it

[@patrickmckinley8739]

A086703. The continued fraction of Levy's constant. Levy's is closely related to Khinchin's. This constant also embodies a property of the continued fraction of almost all numbers - and this sequence is itself a continued fraction. We say "almost all" numbers. Just to expand on this, the exceptions are somewhat intriguing. Any number that is a root of a quadratic does not comply. Also, Euler's number e.

[@CrashSable]

The "almost all" annoys me about most pure mathematicians - it's just conflating quantity with practicality. If I attacked the statement from the opposite direction, I could say "almost all numbers aren't used in any meaningful way, therefore if such a small proportion of numbers are actually useful to us (so small, the percentage has to be rounded 0), then we must conclude that working with numbers at all is useless and all mathematicians should be executed for wasting so many of our resources" Obviously, both statements are equally nonsensical...

[@StefanReich]

7:19 WTH is "base 1"? In the only thing that resembles a "base 1" (unary numbers), 1s are used, not 0s.

[@tribiz6762]

The Poincaré conjecture was proven by at the time a pretty unknown mathematician

[@davidwilkie9551]

Math OCD = Genius, Conjecture.. Nope, Renormalized cynicism between the bounded interval of possibilities is still a personal self-defining assessment by default criteria.

[@mrphlip]

The subtitles for the "metadromes" section constantly says "strict descending" instead of "strict ascending"... I guess the transcriber misheard? But that caused a lot of confusion for me trying to figure out what was going on...

[@farstar31]

That's strange, the link to the video with Tony's favorite sequence is broken, perhaps the video has been taken down or something else?

[@Lightning_Lance]

Yeah, noticed the same. The link in the description works though.

[@MCLooyverse]

The continued fraction of φ wouldn't yield Khinchin's constant, right? Unless you're going to tell me that (1 * 1 * 1 * 1 * ...)^(1/1) != 1

[@whauk]

how do you define the percentage? if you have a finite set you can just count out the number of elements with your property and compute the percentage. if you have a countable set you can look at all finite subsets and count the elements with your property in each of them and figure out whether the "limit" exists if you make them larger. however on an uncountable set...? compare lesbesgue-measures in any bounded subset? and then figure out whether a limit exists if you make the subsets "larger"?

[@keyofamajor]

yes, but when you're dealing with infinities they can't be handled in an intuitive manner such as that. i think of conditionally convergent series and grandi's series as immediate examples that infinity does not obey normal laws. doing so would result in contradiction. you could say that composite numbers are the vast majority of whole numbers by the same logic, but that really doesn't work out.

[@keyofamajor]

i don't think that "almost all" qualifies as "the vast majority" i suppose that irrationals are uncountable while rationals are countable, but rationals and irrationals are both dense in the reals. i'm having trouble agreeing with the brilliance of this number considering that. also, quick note: at 7:24 the place values being whole numbers is erroneous. the place values represent POWERS of the base number, not multiples. i.e. all the places should be labeled "1s" (which is counter-intuitive, ik)

[@BunniBuu]

I would like to point out 1:11 he just says,"We know ___" and leaves it at that. Not everyone knows what continued fraction expansion is. It's not like there was a numberphile video on it. 1:35 "Forget A naught" then doesn't explain why. 2:33 "infinitesimally small set"... in which you can name specific numbers? 4:37 "Golomb invented tetris" would be nice if that was relevant to the video? 5:16 "I couldn't put a 2 here, it doesn't make sense" Why not? No explanation...

[@BunniBuu]

You didn't really explain these very well... you basically just said,"___ doesn't make sense" without really explaining why it didn't make sense. It would help if we knew about what the pattern was trying to do before you showed us how you know what the numbers were... I didn't understand almost any of this because of that >.

[@topilinkala7651]

But as a phycisist one must concur that in nature everything is by rational numbers based on plank's measures. There is not even irrantional numbers in nature. Complex numbers of course but only rational complex numbers.

[@ChristianPerfect]

The important point for a rational is that the top and bottom of the fraction have to be whole numbers. In a circle, at least one of the circumference and the radius is always an irrational number, so it doesn't make pi rational. It's... kind of a circular argument ;)

[@arekkrolak6320]

Andrew Wiles is a genius but I don't think many mathematicians would tell this about him before he came up with the proof. I am not a mathematician myself so I may be incorrect here.

[@Ubeogesh]

How can you hate maths? I mean, it's nothing! Maybe you should look at the most philosophical numberphile's video "do numbers exist?" What I am trying to say is that you can't hate something that you don't even know if it exists...

[@d4m4s74]

I don't know what this sequence is called, if it even has a name, but I like it 1, 11, 21, 1211, 1231, 131221, 132221, 133221, 232221, 134211, 14131221, 14132241, etc. Wether this sequence has a name or not, try to find out why this is a sequence.

[@mac1991seth]

When he said that Khinchin's constant is number that supposedly God would have/take, I knew he would pick this sequence his favourite. All others were interesting but not as much as that one.

[@Promatheos]

Whoever came up with all 7's wasn't even trying. I should submit my all 9's sequence. It's just 9 over and over. What does it mean you ask? I dunno, but the all 7's thing is just as pointless.

[@uraldamasis6887]

I don't understand why Khinichin's constant should be regarded as an integer sequence. We always consider pi or e to be numbers, not integer sequences.

[@uobscdarkside732]

for golombs sequence you could have begun it 211 Edit: I looked it up and the sequence is non descending so 211 wouldnt work

[@Nathanchooper]

I've thought of a cool sequence that no one at my school could get without me telling them. Can anyone get it? 111111211131222232223333334332... This is my favorite sequence!

[@mathmachine4266]

The largest metronome base n is (n^n-n²+n-1)/(n-1)². Special case, n=1, the limit as you go to 1 is 0.

[@Bovineprogrammer]

That's what I was thinking. Don't care for the decimal expansion, but the continued fraction would make much more sense.

[@borntoarun]

Can you do a whole video on Khinchin's constant? Specifically, can you do an example of how a certain number, when you do the continued fraction expansion of it, approaches the constant?

[@ckq]

8 years late but essentially if you pick a random number 0-1, It's continued fraction has a 1/1 - 1/2 = 1/2 chance of being 1, 1/2 -1/3 = 1/6 chance of being 2, 1/12 chance of being 3, etc. So the geometric mean is just 1^(1/2) * 2 ^ (1/6) * 3^(1/12)... n^(1/(n(n+1)) which is that constant

[@d4m4s74]

I see I made a mistake: The full sequence is 1, 11, 21, 1211, 1231, 131221, 132231, 232221, 134211, 14131231, 14231241, 24132231, 14233221.

[@NikoRonkainen]

2:53 if god had a number, it would be 20 aka god's number. Every position of the Rubik's cube can be solved in 20 moves (HTM) or less.

[@KINGOFAERODRONE]

Hi my friend told me that if you keep halving a number you will reach 0 , do you have any videos on this?, I am not sure if he is right.

[@TheBalfrog]

Wieferich Primes are hard to explain, but the best I can give it to you is by simply showing it, Wieferich Primes we know of, 1093, so p=1093, 2^(p-1) which is 2^(1092) can be divided by 1093, and come out with an integer, whereas if you tried say p=5, (2^4)/5 isn't an integer. Because you can rewrite the conjecture 2^(p-1) = 1, it needs to come out with an integer, to be a Wieferich Prime, hope that sort of helped with understanding it

[@EbbeTheGuitarist]

That's because the circumference can't really be written as fraction because it's made up by 2*pi*radius, and pi is irrational. In schoolmath you always get simplifications like circumference = 10, but it's really an irrational consepts. In real life applications we round it up, but it's rally just an irrational numer * rational numer = a new irrational numer. If you use like 50 desimals of pi, then you could messure the circumference of the universe by only the lenght of a hydrogen atom wrong!

[@dwarduk2]

Take pi for example. a0 is the integer part of that, so 3. Now take the reciprocal of the fractional part. a1 is the integer part of that, so a1 = 7, giving a geometric mean so far of 4.58. Take the reciprocal of the fractional part of what you currently have, and a2 is the integer part of that (15), and so on. I'm going to reply to this comment with some actual data on this as it applies to pi.

[@burk314]

I agree. Khinchin's constant assumes that the aliens would have even invented continued fractions in the first place. While I do find them interesting, continued fractions are kind of a niche concept in number theory and aren't really important to a lot of mathematics. Then Khinchin's constant is a non-obvious derivation from that which doesn't work for common numbers. Compared to pi which pops up in many different areas of mathematics, Khinchin's constant is not that important.

[@burk314]

With the composite numbers vs the primes you'd be comparing two countably infinite sets. You need another way to compare the sets, say natural density in which the primes have density zero. So in that sense, yes the composite numbers are the vast majority. You are correct in that we need to be careful about our definitions and "vast majority" is vague, but the phrase 'almost every' in mathematics, oddly enough, has a very precise definition which has little to do with topological density.

[@modus_ponens]

So a₁ a₂ a₃ ... are just numbers that fit to construct that number in that specific way? It should be said in clearer way that this applies only for irrational numbers. But according to wikipedia √19 repeats with a perioid of 6. So ...hmmmphh... ((2*1*3*1*2*8)¹⁰⁰⁰)⁽¹⁻⁶⁰⁰⁰⁾ ... (sorry no up index slash with sift+^ + /)... =2,14... so this did get somewhat close, but it's no near 2.69... It is maybe just my computers calculator or this is one of the numbers that doesn't fit.

[@ExtremeMagneticPower]

A continued fraction expansion is basically a decimal turned into a sequence, and any digit is turned into an integer. Let's take pi as an example. 3 is the first digit, so that will be the first number in the expansion sequence: a0. 1 is the first decimal: a1 4 is the second: a2 The sequence would be: {3,1,4,1,5,9, ... } In the above sequence, if I was to take a0 + 1 / a1, it would give me 3.1 If I was to take a0 + 1/ (a1 + 1 / a2), it would give me 3.14 Make sense now?

[@Boomshicleafaunda]

Uh, better example. Think back to 1st Grade when you learned Place Value. The 2 in the TENS place means 2 tens, or 20. In base 8, a 7 in the EIGHTS place means 7 eights, or 56. And where we have the 100th Place, or 10^2, in base eight the second place would be 64th, or 8^2. Thus replacing the 1,000th Place, or 10^3, with 512th Place, or 8^3. Example: 123 [10] = 123; 123 [8] = 83; 123 [5] = 38. As your example stated, numbers in base [b] go from 0 to b-1, and after 9 we tend to use letters.

[@ehhorvath13]

in a nut shell, a base is how you describe the place holder. we generally use base 10, so each place holder is a power of 10: one's place, 10's place, 100's place...or in other words 10^0 place, 10^1 place, 10^2 place. If I want base 5, each place holder would be a power of 5. 5^0, 5^1, 5^2 ect. so if I want to write the number "six" in base 5, I would write 11. meaning, one set of 5^1=5 and one set of 5^0=1 ==> 5+1=6.

[@zelda12346]

People can rattle off the first few digits of Pi and e, but most people define Pi as the ratio between the diameter of a circle and its circumference...which requires you to measure exactly since in order to calculate one of those things, you need Pi. However, there are other ways to exactly calculate Pi and e via the sum of an infinite series. I would like to see the proof as to why these work

[@a.gorlovicha9169]

Isn't 3 a Wieferich prime? Because 3^2=9 and 9 divides by 3 (2^3-1)-1

[@ThorHC11]

Видео от A.Gorlovich'a You've got it backwards. it's that p² divides 2^(p-1)-1, not divides by it. for three to work, 3 would have to divide by 9, and it doesn't, because 1/3 is not whole.

[@razzamattazz]

If it began with 0 then I'm pretty sure it would have to just be an infinite sequence of zeros. 0, 0, 0, 0, 0, 0, etc. How many times does 1 appear in the sequence? Well, the 1st term is 0 so it has to appear 0 times. How many times does 2 appear in the sequence? Well, the 2nd term is 0 so it appears 0 times, and so on and so forth

[@anticorncob6]

Of course pi can be written as a fraction, ALL numbers can. Irrational numbers cannot be written as a fraction where BOTH NUMBERS ARE INTEGERS. This means it is impossible for there to exist a circle in plane space where the diameter and circumference are both integer lengths, no matter what unit you use.

[@joebunny3807]

It's not intuitive, so let's take a series that depends on the number of digital rays of the forelimbs that some animal species on some planet *happened* to have when inventing their number system. That's not how you capture the spirit of that "universality" that Khintchine's constant is all about. So Elliott couldn't have said it better: it was exactly the wrong choice.

[@MultiPaulinator]

Actually, those are just the powers of 2. A perfect number is a number whose entries in its divisor list -- including 1 but not the number itself -- add up to the number in question. Finding them goes something like this: Iff(sic) 2^(p-1) is prime -- which is only possible, though not guaranteed, when p is prime -- then 2^(p-1)*(2^p-1) is perfect.

[@christophersam1989]

Here he means 'almost all' in the measure theoretic sense, rather than cardinality. Just as the interval [0,1] contains 'almost none' of the numbers in the Cantor set, despite being an uncountable subset. Essentially if you picked a number at random there is probability 1 that it gives Khinchin's constant and probability 0 that it lies in the Cantor set.

[@Sharaton]

You've also missed that there are n factors in the product. If they all were equal to x the product would be x^n and the exponents would cancel out. When you take limits you have to take the limit of the entire expression, not just parts. (Furthermore, the limit would be of the type infinity^0 which is undefined)

[@TeamDragofied]

does khinchin's constant work with -1/12?

[@Cernoise]

The Meem no, because that's a rational number and it doesn't work with rational numbers.

[@TeamDragofied]

Angela Brett it was a joke just look up “negative one twelfth”

[@Cernoise]

I have seen the videos about -1/12, but didn't really see why it would be funny to mention it here so I thought you might actually think it was a special case or something.

[@TeamDragofied]

Angela Brett I guess it’s just kind of a meme here because there’s so many videos with a mention of it

[@Cernoise]

I guess it's like the Parker Square of numbers, then.

[@redfayl]

It's pretty difficult to do with any number because it does not work with rational numbers. It might seem that is not the case because he says it works for almost all numbers, but the fact remains that there are infinitely more irrational and trascendental numbers than rational ones.

[@redfayl]

It doesn't work for rational numbers. So x being 3 does not work since a0 would just be 3. Divides into just means it is a factor of. Or in more mathematical terms, n divides x if x is congruent to 0 mod n (leaves no residue once you do the division algorithm.

[@anticorncob6]

I used to debate religious people a lot, don't do it anymore unless they bring it up first. And yes, I do know shockofgod and the Hovinds, but not the other people you mentioned. You're right, they are religious nuts, and probably bigger than factsvs science.

[@doug65536]

I notice some complaints that this video has more advanced content than usual - and that's ok! I'm seeing similar comments in the network stack video in the computerphile channel. Please, let some videos be more advanced, so more advanced viewers can have some enjoyable content too.

[@einsteiner900]

Read the Wikipedia article on continued fractions. All rational numbers have terminating (non-infinite) continued fraction representations. Therefore the geometric mean of their terms does not "approach" anything, it just is a fixed value, which will not be the same as Kinschine's constant.

[@Beer_Dad1975]

I agree, I had to watch it a couple of times, do a google search and work through on a piece of paper myself to understand the first two at all - that's the first time that has happened ever with a video on this channel. Admittedly I'm not the brightest stump in the forest, but I'm no idiot either.

[@burk314]

In that video, James is talking about natural density in the integers which is a different concept than the measure theory definition of 'almost every'. (In fact James shouldn't have used 'almost every' in that video as natural density is not really a measure)

[@rkubiniec]

Giving that constant to aliens not only assumes they understand whatever numeric system for transmitting the data we use, but it also assumes that they do math the same way as us, have found the constant, and/or would be able to derive it once we've given it to them. That seems really unlikely.