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.

@user-mz7cn9hq8v

4 жыл бұрын

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

@aforcemorepowerful

2 жыл бұрын

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

@TheMagicianLiam

2 жыл бұрын

I agree but in base 12. Sorry

@hkayakh

2 жыл бұрын

How about that sequence but nth position is -n?

@sankang9425

Жыл бұрын

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

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

@leo17921

4 жыл бұрын

also cause its more complicated

@rednecktash

4 жыл бұрын

thats what i thought too even before seeing any other ones

@thomaskaldahl196

3 жыл бұрын

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

@olivialuv1

Жыл бұрын

@@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

Жыл бұрын

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

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.

*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

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

2 жыл бұрын

Wow!

@bagelnine9

7 ай бұрын

Same.

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

2 жыл бұрын

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

11 ай бұрын

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

Shout-out to Ireland!

@goutamboppana961

2 жыл бұрын

??????????

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.

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

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

for the wild numbers, just add 0.5.

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

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

@IMortage

4 жыл бұрын

Numberphile had Conway himself talking about the look and say sequence

@ckmym

4 жыл бұрын

1 11 21 1211 111221 312211 13112221 1113213211 ...

This is the only Numberphile video that went completely over my head.

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

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.

I already knew about khnichin's constant and love it, but golomb's sequence is definitely my favourite!

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

2 жыл бұрын

You were ahead of your time...

@asheep7797

2 ай бұрын

4 years ahead of your time.

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

@annevanderbijl3510

3 жыл бұрын

Nice

Im watching the entire series this summer and i cant stop thinking of this lol

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.

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.

Solomon Golomb ! That is a great name.

Sick sequences.

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.

fun video. I really liked it and tony is a good at explaining.

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

More often then not this channel does a good job at explaining the math so that I can understand it and how cool it is even though with my basic knowledge. This is not one of those times. I am sure it is awesome, but it is way over my head.

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

You can't beat a bit of numberphile during the school summer holidays , especially in ireland where it always rains !

Great work numberphile!!!

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.

I knew Khinchin's constant would be his favourite. That sort of kinky stuff makes all mathematicians salivate.

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?

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

@mr.z111

Жыл бұрын

Ш vs Щ

@gmdFrame

11 ай бұрын

@@mr.z111 Прив

Golomb's sequence actually kinda blows my mind.

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

YES! I use oeis all the time! :D Awesome that you guys use it too

yep khinchin's would definitely be my favorite of those

My favourite: 1, 2, 6, 12, 60, 360 and 2520. The only numbers that have more divisors than every single number apart from itself and up to it's double. These are literally the most divisible numbers can be, seeing as doubling the number adds a new power of two to the factors.

@skalderman

9 ай бұрын

is 2520 the maximum how about 5040?

@robo3007

9 ай бұрын

@@skalderman 5040 doesn't work. 7560 has more divisors than it and is less than 10080 (2*5040)

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

I noticed what Tony's favourite sequence was from how he spoke of it. =)

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.

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.

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

@user-kh5tv9rb6y

5 жыл бұрын

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

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.

Thank you for the great answer. I'll be hard-pressed to find notebooks of brown, quadrille paper (my go-to as an Mech. Engineering major), but maybe I'll try it out sometime.

that grows exactly as fast as the busybeaver function. you could however use f(x) = busybeaver(x) * busybeaver(x)

This was explained very well.

They've used it for a while (since the beginning I think). It's provided by Brady, the person who runs the channel and films the videos.

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

2 жыл бұрын

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

I think the interesting thing about Khichin's number is that it neatly avoids the rationals.

@vocnus

2 жыл бұрын

Yes, of course... and there is even more interesting about it, especially if we start it with pi.

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

Infinite fraction is a decimal rotation of digits. As the fraction increase the decimals are insignificant and so reduce to k constant. Two most significant and other reduce fast. 3 is the closest. These kind of things are wave guides. Mostly used for encryption FM and AM.

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.

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.

YES! Well done.

That makes absolute sense to me now. Thank you for explaining the concept. On another note, would you know why some people argue that base 12 is more intuitive than base 10?

@tomlloyd8122

2 жыл бұрын

12 has more factors

i like this video keep it up bradey

AH! I understand now. Thank you. Just to make sure then: in base 3, would a number such as 5432 be 543 groups of 3 + 2, or 1631? (and it would continue like that?)

I love math... I also love the nerdiness in all of these videos!

Would it be possible to see a video on Golumb's Ruler? I can see some clear musical applications, but I'd like to see it from a mathematician's perspective.

Love it!

All the seven's is my favorite.

There must be some other exceptions to Khinchin's Constant than just rational numbers. For instance, the Golden Ratio is an irrational number, and the continued fraction expansion for it is an endless series of 1's as the coefficients, meaning that the geometric mean of the coefficients would be 1 for any number of iterations, rather than converging to the Constant as the number of terms in the expansion approaches infinity. Similarly, any irrational number constructed in the same way so that all the coefficients in the fraction expansion have the same value would have a geometric mean equal to that value instead of the the Constant.

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?

Please do a video on the look-and-say sequence! You start off with some seed like 1, then you say it out loud: "there is one one (1 1)," and so the next term is 11. Then you do it again: "There are two ones," so the next is 21. And then "one two and one one": 1211. It has a lot of unexpected properties and is just downright cool B)

The problem with ten is that it uses the factors 2 and 5 when using 2 and 3 would give you just as many divisors, but you would be able to use those factors more frequently as the factors occur in other numbers more frequently. So every benefit you get from base ten you would get from base six, but they would occur more frequently. There are infinitely many numbers you can choose from that have a given number of divisors, but the smaller ones give you the benefits of those number more often.

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)

Brady, please make a video about e! I'd love to see it.

Oh, of course. that makes perfect sense. Thank you!

Will Khinchin's Constant also work for complex numbers? Or at least their real parts or values..?

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.

A similar sequence would be all p^3^n where p is prime and n is a non-negative integer: 2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, ... You can get every positive rational number excluding 1 by multiplying or dividing these numbers and factorisations are unique. For example, 28/5 = 7 * 8 / 2 / 5

you have a nice office, even a conservatory, nice !

numberphile is an awesome channel!!!!

Khinchin's constant is gotten by writing numbers in a specific way (continued fraction) - are there other way's of writing numbers, which beget other constants?

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.

I can't follow some of these videos, but it's knowledge so.....keep it up! Also I want more :D

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

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

@XenophonSoulis

4 жыл бұрын

312211 RIP John Conway

@evanknowles4780

4 жыл бұрын

13112221

@manmanman4825

4 жыл бұрын

@@evanknowles4780 1113213211

the forth; take any number, write it out, count the letters for it, write that number out, repeat, 4. dunno what you'd call it but i like it ^^ and it works in english, german, dutch and probably some other languages

Andrew Wiles proof of Fermat's Last Theorem takes a genius to even understand it. And he used some math that didn't exist in Fermat's time. It has yet to be truly solved in the way that Fermat first thought of.

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.

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

he wrote the general form of a continued fraction expansion. a0, a1, a2 are the digits of that expansion.

ah, i see. thanks for clearing it up. it's pretty clear that only irrational numbers result in khinchin's constant. and rationals are dense in the reals. so it's a little hard to wrap your head around, but i never expected for it to be something intuitive, i suppose.

Imagine the number of the base was the "ten", so, for example, in base 10 you got 123, to calculate that you got 1*10^2 + 2*10^1 + 3*10^0 = 123, you can do the same in other bases like 123 in base 4 is 1*4^2 + 2*4^1 + 3*4^0 in base 10

Thumbs up if you could already feel Tony's excitement when describing the Khintchine's constant :)

In the way the Fibonacci Sequence has values by summing the previous 2 values, do any constants or behaviors surface by increasing the number to 3 or higher? 1, 1, 1, 3, 5, 9, 17, 31...

Could you explane Where the simpons factor was calculated, and how ? Such as the factors 1,4 2,4,2,4,1. Or depending of how many factors you need

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

@earfolds

9 жыл бұрын

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

@CoyMcBob

9 жыл бұрын

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

9 жыл бұрын

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

9 жыл бұрын

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

9 жыл бұрын

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.

i think it would be interresting if you made a video about fermats last theorem

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.

OEIS A002211 (the continued fraction for Khintchin's Constant) beats A002210 every time. As a bonus, its OEIS number consists solely of repeated digits.

I'm sure for some of the sequences if he expanded on the explanation then it would be understandable. Fitting that many (i can imagine) complex mathematical sequences with full explanation in a ~14 minute video is near impossible. Good job though and great video. Gives you a lot to think about either way.

He really failed at keeping a "poker face", haha. Anyway, I think my favorite of these would be Golomb's Sequence. I was skeptical when i saw it first, but I was simply blown away when the Golden Ratio showed up :p

i still can't get the Golomb's sequence

Can someone explain how Golomb's sequence moves towards the Golden Ratio as n approaches infinity?

Which is why they should do a video about it. Every one knows about pi too and they have lots of videos about it.