Random Fibonacci Numbers - Numberphile

Dr James Grime on random Fibonacci Sequences...
Extra footage: • Random Fibonacci Numbe...
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Fibonacci Numbers in the Mandelbrot Set: • Fibonacci Numbers hidd...
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[@numberphile]

Extra footage: kzbin.info/www/bejne/fGGmZYhthpqsjrc Fibonacci Numbers in the Mandelbrot Set: kzbin.info/www/bejne/an20p52Jm5uGidk More James Grime videos: bit.ly/grimevideos

[@TheRealGuywithoutaMustache]

I find your videos really fascinating

[@prunabluepepper]

Welcome back Dr. Grime. Hope you're well. Have a great start into the coming week :)

[@aswinkumar6052]

Bro I got an idea if any prime number squared itself and subtract 2 can be a prime. (Prime)^2-2 =may be prime

[@Zoxesyr]

which video did that framed brown paper come from?

[@robertlozyniak3661]

How about a link to the paper?

[@rohansastry]

Every year, I throw a fibonacci party. Each party is as big as the last two combined.

[@geomochi4904]

Rohan Sastry lol

[@miramosa7768]

I do the same, but the first two had zero attendees so I'm not getting anywhere :(

[@randomdude9135]

@@miramosa7768 You take the factorial of them and proceed ;)

[@TuberTugger]

@@miramosa7768 You didn't show up to your own party?

[@chenugent]

XDDD

[@ZachGatesHere]

I'm a simple man. I see James on Numberphile, I'm in.

[@NightBlado]

Came scrolling here for this

[@galacticbob1]

I see a new Numberphile video and I'm N 😋

[@TheExoplanetsChannel]

Me too

[@wingsofmathematicsbytanush2507]

@@TheExoplanetsChannel me to also.

[@obaroajeke6080]

I signed in to like this comment

[@Justuy]

And I am wondering why he isn't calling it 'RANDOMACCI NUMBERS'!

[@harishchalwadi]

James Prime is back 🙂

[@lindhe]

He's in his prime.

[@TheExoplanetsChannel]

Yay

[@maxonmendel5757]

yeah what had happened with that?

[@DanDart]

Prime is my ship name for Parker + Grime. It's the perfect mathematical ship (of Theseus).

[@Triantalex]

false.

[@johnchessant3012]

Fun fact: The author of the paper at 9:20 discovered a proof that there are infinitely many primes using basic topology.

[@noidea2568]

A proof that there are infinitely many primes ysing basic TOPOLOGY? I need to see this!

[@spyfox260]

No Idea Yes, a proof that there are infinitely many primes using basic topology. You are correct

[@johnchessant3012]

No Idea Yes! Look up "Furstenberg's proof" on Wikipedia. You'll need to know the definition of a topology.

[@yusuf-5531]

@@johnchessant3012 is there a proof that there are infinitely many primes using basic topology?

[@jingermcblabbersnitch7162]

Read it. Neat

[@rattyoman]

the way he said "do you wanna hear about applications of fibonacci sequences?" at the end was so precious

[@banjofries]

I was about to say something funny about this but then I tried it, and it's actually impossible for the sequence to have 2 zeroes in a row, because there will always be at least a 1, 2, -1, or -2 somewhere behind or in front of the zero due to zero not being able to change the value of the previous number in the list to zero since the sequence starts with 1. Oh well, I guess there CAN'T be a list that's a few numbers then just infinite zeroes...

[@alvarol.martinez5230]

a slightly higher level explanation of this: the vector (a_n, a_(n+1)) is the product of a 2x2 matrix A = (0, 1; +1 or -1, 1) by the nonzero vector (a_(n-1), a_n). Since A is invertible, the result can never be the zero vector

[@mirabilis]

Unless it's all zeroes.

[@mirabilis]

Easier: Two zeroes in a row would mean the number before those were a zero and the number before that was a zero and so on...

[@Shenron557]

@@alvarol.martinez5230 Wow, beautiful explanation. Thanks!

[@BlazingshadeLetsPlay]

Álvaro L. Martínez as someone who is just learning linear algebra rn my mind is blown. I thought matrix algebra was just a boring strategy to compute stuff but once again math blows my mind

[@Gooberpatrol66]

9:53 I love that they defined an IEEE double float in a mathematical paper

[@otakuribo]

Audrey: 🐶 *casually enters room James, excitedly: *Do you want to hear about applications of the Fibonacci sequence?*

[@TheExoplanetsChannel]

.

[@Triantalex]

??

[@maulob1523]

"Nobody will make a computer simulation." I'll take that as a challenge. I got some results: I did 45000 random sequences, and ended up with 0,98721748 It seems like 64 bits are not enough for enough terms, and enough precision for calculating averages And I can't be bothered to code more. BUT! While doing that I figured out some cool identities: ( F(2^k) )² + ( F(2^k+1) )² = F(2^(k+1)+1) F(2^k) * (2*F(2^k+1) - F(2^k)) = F(2^(k+1)) Example: (F9)²+(F8)²=F17; F8*(2*F9-F8)=F16 Where Fn is the nth Fibonacci number, of course

[@giladkay3761]

how did you do the program, I was going to start when I realized the second I try to find the value of something divided by 0 the program will crash

[@DehimVerveen]

@@giladkay3761 Why would you divide by zero? You can use the nth root. Also, you can check if the value you're dividing by is zero before doing so.

[@tinydong4586]

Is it the nth term, nnd term or nst term?

[@DehimVerveen]

@MauLob, Would you reckon 128 bits are enough? How many bits do you think are necessary?

[@maulob1523]

@@DehimVerveen I do not know. With 64 bits you can, worst case, only get to about 90 terms. With 128 terms you could do a little more than double. For the normal fibonacci sequence, 90 terms is enough to hit the golden ratio with double precision. However, I am starting to think that the problem is not integer over/underflow, but floating point precision. And at that point, I have absolutely no clue.

[@albinoagellar268]

James on thumbnail = instant click

[@Reneofficialid]

Me tooooooooo! Hahaha

[@Rickety3263]

Yaaasss😂 Why do I find this so fascinating 😅

[@Martin-qb2mw]

Agreed

[@TheExoplanetsChannel]

Yes

[@Triantalex]

??

[@bhooshanpandit1344]

Yes!!! Please *MORE JAMES GRIME* & *MATT PARKER* & *TONY PADILLA* !!!

[@gabor6259]

*& Hannah Fry & Tadashi Tokieda & Cliff Stoll & Holly Krieger & Simon Pampena & Zvezdalina Stankova & Ben Sparks & ...*

[@TheExoplanetsChannel]

Yes!

[@fmakofmako]

Surprised James didnt say anything about the 1/sqrt(5) in the first few minutes. It would have made his approximation extremely accurate.

[@kaaiplayspiano7200]

(1+sqrt(5))/2*

[@fmakofmako]

@@kaaiplayspiano7200 probably I should have clarified, because I assumed everyone knew this. The formula for large n for fibonacci is phi^n/sqrt(5). If you watched the first few minutes of the video, then there is a moment when he takes Phi to a power and compares it to the actual value and then hand waves the differences although notes the magnitude is the same. That's what would have been a lot closer if he divided by root 5.

[@ashtonhoward5582]

@@fmakofmako it took me a bit to figure out what phi was. Also, for that you need to round to the nearest integer.

[@luisliquete9027]

fmakofmako it works perfectly!! Now, im wondering if the 5 from sqrt(5) may have any relation with the golden ratio that appears in PENTAgons (i dont remember in which exact video i saw this). I might be setting my expectations a bit too high on this one, but who knows??!! 🧐🧐

[@mastod0n1]

@@luisliquete9027 I assume the square root of 5 is coming from the fraction representation of the golden ratio, which (1+sqrt(5))/2

[@silverette666]

i haven't checked this channel in a long while (a few years) and i'm glad to see james is still doing videos with you. he was my favorite part of your channel back when i watched your videos regularly

[@leonhardeuler9839]

I see the word “random” and James, I click instantly.

[@danfg7215]

it had me at Fibonacci

[@tomaszkantoch4426]

Why. James is not just a random guy :D

[@starsian]

I see Euler, I like instantly.

[@leonhardeuler9839]

Jay N Who?

[@leadnitrate2194]

Jay N but not as proficient.

[@shugaroony]

This vid is just like classic Numberphile. James Grime, brown paper, and no fancy graphics with silly sounds. Thumbs up. :)

[@ragnkja]

The “almost surely” looks a lot like the “almost all” in the video about how almost all numbers contain the digit 3.

[@karapuzo1]

That was some exemplary handwaving, I hope the original paper is more rigorous.

[@calculator7774]

@@karapuzo1 Actually, something happening "almost surely" is a rigorous mathematical phrase. It means the odds of it happening are 100%. Even though it is somewhat counterintuitive, this is not the same as saying that it will always happen, hence the "almost".

[@karapuzo1]

@@calculator7774 Yes, I am aware. This still requires rigorous proof that the probability of the examples where the ratio does not converge is 0.

[@OlliWilkman]

"Almost all" in terms of real numbers means that the set of counterexamples have a measure of zero. Since probability theory is usually formulated in terms of measure theory too, I suspect the analogy is that "almost surely" means that the probability measure of the inverse statement approaches zero?

[@dlevi67]

@@OlliWilkman If the cases in which it does not converge to Viswanath's constant are those and only those with patterns, those can be equated to rational numbers in binary, which have measure 0 in the set of the reals ("all" random binary digit numbers between 0 and 1)

[@someoneunknown6553]

I've been binge watching the James Grime playlist

[@Exeedo.]

What I have been thinking while watching this video is: what would happen if the probability between getting a plus or a minus is not 50:50? I think that will be interesting to find a formula/algorithm to find what number the growth rate approach depending on the probability.

[@123amsterdan456]

Love the Grahams number paper on the wall :)

[@ItachiUchiha-ns1il]

I’m curious as to whether or not the growth rate is transcendental or algebraic for the random Fibonacci.

[@randomdude9135]

Divakar Viswanath It's rare to see a Mathematician with an Indian name on Numberphile. And I'm happy to know about his finding 😊

[@MrEuller88]

You mean any Indian mathematician besides Ramanujan. That guy really made his impression in the mathematical world.

[@erickcapitanio1957]

and an awesome sounding name on top of that

[@harikishanrakhade6108]

And what about Kaprekar who gave the Kaprekar constant?

[@DarkMage2k]

@@harikishanrakhade6108 alright calm down this isn't a contest

[@davidgillies620]

Off the top of my head I can think of Bose (Bose-Einstein distribution), Chandrashekar (white dwarfs), Varadarajan (supersymmetry) and Agrawal (AKS primality test). India is rightly celebrated for the calibre of mathematicians it turns out.

[@danieloh6782]

Was just watching James Grime! Always look forward to his videos :>

[@senororlando2]

Damn Dr. Jimmy hasn’t aged a day in 10 years

[@Tehom1]

For a moment I thought there was some likelihood of getting a tail of all 0s. Because James only needed to hit 0 once more, and after that he's always adding or subtracting two 0s and getting 0. But on reflection, that's impossible. Once you have a nonzero X followed by a 0, the next one is either X or -X and never 0.

[@shadowhejhog]

oh that’s fascinating! i thought it would end up as 0,0,0,... too.

[@thej3799]

You're moving into the ratio of what defines a dimension

[@flikkie72]

To get a (random) sequence of this in excel: Put 1s in cells A1 and A2 and put "=A1+A2*(-1)^round(rand(),0)" in A3 and pull it down.

[@johnchessant3012]

Fun fact: The nth Fibonacci number is given exactly by rounding (phi)^n / sqrt(5).

[@shambosaha9727]

Ah, yes. Of course you can exactly figure out what the answer almost is.

[@andywright8803]

@@shambosaha9727 no, you can figure out what it is EXACTLY. It's (phi)^n / sqrt(5) rounded to nearest whole number

[@shambosaha9727]

@@andywright8803 But you're still rounding

[@CauchyIntegralFormula]

Yeah, but (phi)^n/sqrt(5) isn't exactly F_n. The closest integer to (phi)^n/sqrt(5) is exactly F_n

[@shambosaha9727]

@@CauchyIntegralFormula That's what I said.

[@sheerrmaan]

I love his passion and his way of communicate things

[@kasperjoonatan6014]

That is the most beautiful shirt he has ever had 😮

[@TXKurt]

To those writing programs: Look at the expression at time 6:45. |R_n|^(1/n). I believe this is what you want to average. By averaging this over many random series, already with n=40 you should get around 1.125. Going up to n=1000, the result is starting to look familiar: 1.13174.. (400000 series averaged).

[@yayaaabunni]

I swear if JG was my math professor, I would be front row, every clad and turn in every assignment.

[@Danicker]

I love how "almost surely" is a techincal mathematical expression. Great vid as usual!

[@AaronRotenberg]

7:45 is a little misleading. "Almost surely" means the probability is _exactly_ 100%, not "almost" 100%. The catch is that, for infinite sequences of events, 100% probability ≠ guaranteed.

[@ragnkja]

Aaron Rotenberg This is explained very well in “3 is everywhere”, one of the first videos on this channel.

[@flexico64]

I was thinking of the explanation involving a dart board that explained some things are "possible but have 0% probability." *brain starts smoking*

[@alainrogez8485]

0:08 "We have to recap the Fibonacci sequence first". Didn't Numberphile do hundred of videos about this sequence, did it?

[@ragnkja]

Alain Rogez Just in case this is someone’s first Numberphile video about Fibonacci numbers.

[@Qbe_Root]

Each Numberphile video about the Fibonacci sequence is equal to the sum of the previous two

[@martinbergman7693]

Are those framed pictures in the background (which has been standing on the floor for many years, it seems) ever going to be hung on a wall? That's what I want to know.

[@robin888official]

You have to divide phi^1000000 by sqrt(5) to get the 1000000th fibonacci number. (By an error of only 0.618^1000000, so the formula gets extremely accurate!)

[@ianflanagan209]

1, i, 1+i, 1+2i, 2+3i, 3+5i, 5+8i....creates 2 fib sequences simultaneously as the real and imaginary parts follow the fib sequence. this could be seen as the sum of 2 separate fib sequences, fib real+fib imaginary which is (f_n+1)+(f_n+1)i=((f_n-1)+(f_n)+(i*f_n+1)+(i*f_n)). The cool thing about this is we get a+bi format which can be represented as re^i*t and the limit of the respective sequences is phi and i*phi so we have a system that combine 3 of the most important constants in math phi, e and i.

[@LelouchLothric]

James is so amazing!

[@flytoheights1]

Love your videos as always!

[@scottanderson8167]

Yay!! A Grimy video!! Love James Grimes!

[@GoingsOn]

This Viswanath’s constant kind of reminds me of Mills’ constant θ, being hard to calculate and also being based on integer sequences.

[@delores1656]

I could listen to James explain anything for hours.

[@johannesh7610]

The fibonacci number Fn = (φ^n + φ'^n) /sqrt(5), where φ/φ' = (1 ± sqrt(5))/2. This is the exact formula

[@sam111880]

You can also apply this random fib sequences to one dimensional random walks of particular step patterns rules. Pretty cool stuff 👍

[@nO_d3N1AL]

It's amazing how quickly Fibonacci tends towards the Golden Ratio: 1, 2, 1.5, 1.667, 1.6, 1.625, 1.615...

[@KevsCoolProductions]

google binet's formula if you wanna see why

[@starsian]

yes, because the growth is exponential

[@aldobernaltvbernal8745]

pick 2 random numbers and use them as your starting point, then do the same thing as you would do in the fibonacci sequence, it will still converge to the golden ratio.

[@zanedobler]

It actually doesn't approximate it all that quickly. The golden ratio has the simplest continued fraction, making it the most irrational number.

[@romekhanna]

He's back... Love his passion

[@triskel20]

"And thats as far as we got" - - Amazing ending, for a moment I thought he was going to say they discovered hundreds more digits!

[@geoffroymb]

I wish James Grime was my math teacher at school.

[@yogipro183]

Fibonacci series is also known as Hemachandra series in India since Hemachandra proposed this series very much earlier with fantastic application in music and architecture of statues.

[@kevina5337]

Good to see Dr. Grime is still alive... haven't seen him in a while lol

[@laojackos]

James Grime yessssss

[@kyraaa__]

I’m going to code this :D

[@numberphile]

Give us some examples of what you get for n=1999,999 and 200,000 and how close it is to the constant!?

[@bananoramatfw]

@@gonzalogarcia6517 bro what are you on about

[@Maniclout]

@@gonzalogarcia6517 * visible confusion *

[@bananoramatfw]

@@gonzalogarcia6517 this comment is pristine

[@bhooshanpandit1344]

**shook**

[@danuttall]

9:30 "Wouldn't the pluses and minuses just cancel out over the long run? No they don't." Cosmologists have been asking that question too, when it comes to matter and antimatter. Because they don't quite cancel out, we have lots more matter in the universe than antimatter.

[@tkarmadragon]

I used to code all sorts of fun graphics simulations with fibonnacci when learning python. You can print out very complex patterns and shapes by applying fib sequence or phi in various ways. For ex. combining fib with modulo operators or Egyptian fractions. I'm not a mathematician so I don't know the theory, but I would draw out many strange patterns within Fibonacci using computer graphics.

[@Veptis]

So any binary number can be made into one of these sequences. And each number has a growth ratio (if you take the rear most element or the moving average). That means you can map integers to real numbers? But you can't as every integer has a limited number of binary digit so it's not an infinite sequence. But if you inverse numbers, you can map reals to reals using this and find some interesting bits of bijictives.

[@TheKemalozgur]

2:20 Actually you can increase the accuracy dramatically by not using f(1)*g^1000000. Because obviously in the first ones golden ratio is nearly meaningless. Lets say, starting from fifth fibo; f(5)*g^999995. Result becomes 1.9691 x 10^208987, which is very close to actual. Or; f(10)*g^999990 ~= 1.9531 x 10^208987.

[@jerryiuliano871]

A close approximation to the Viswanath constant is ln(500*pi)/ln666 = 1.1319812464

[@aaaichunder]

Apparently i am the thirteenth viewer.... I wonder where the 1st, 1st, 2nd, 3rd, 5th and 8th is?

[@randomdude9135]

7290 here

[@Tehom1]

Because having two people who are "first!" is just normal.

[@leadnitrate2194]

Tehom it really is, though with the number of people exclaiming “First” in comment sections these days.

[@TheExoplanetsChannel]

:O

[@VidNudistKid]

Your comment had 89 likes before I got to it

[@orange-micro-fiber9740]

Feels similar to a 1D random walk, but random walks are usually just 1 unit.

[@GerSHAK]

Easily the most mindblowing fact in this video to me is that 1999 was twenty years ago...

[@anteaters-R-us]

Yay!!!! Hi Dr James, great video

[@jillkitten5388]

Wish they would have more formally mentioned: Fib(n) = Round(Phi ^ n / Sqr(5)) which gives the nth Fibonacci number.

[@bobbycraig2583]

round() is a python function.

[@jillkitten5388]

@@bobbycraig2583 It is not just a python function, it is a function in almost all programming languages [in one form or another], the point is that it is the common round function/procedure which in text is hard to represent the mathematical symbolism in an unambiguous way, so by representing it as a function called "Round()" is the simplest most unambiguous way to represent it.

[@bobbycraig2583]

@@jillkitten5388 i know but i only know python. i use [ ] to show rounding

[@rupertmillard]

Great to see the enthusiasm. The problem is captivating. I would have liked more detail about the proof but maybe it’s just too hard!

[@nin10dorox]

2:23 The approximation with F can be a lot more accurate. Phi^x becomes proportional to the golden ratio, but doesnt approach it. If you multiply by (phi + 2) / 5, the approximation will be incredibly close.

[@gresach]

The difference between phi^1,000,000 and F_1,000,000 was glossed as some kind of rounding error. But the ration is sqrt(5), because F_n = (phi^n - psi^n)/sqrt(5). Since psi = (1-sqrt(5))/2, it is transient, as you take it to bigger and bigger powers it comes increasingly close to zero. So psi^n/sqrt(5) will become closer and closer to being a whole number, namely F_n.

[@SimonTiger]

No. phi^1000000 gives the 1000000th _lucas number,_ not fibonacci number. In order to get the fibonacci number, you have to divide by root 5.

[@antoniodagostino5891]

Your comment isn't completely correct phi^1000000 isn't equal to the 1000000th Lucas number, it gives the 1000001th Lucas number instead. 1.618 x 1.618 x 1.618 x 1.618 for example, doesn't give as result the fourth Lucas number, but the fifth. So in order to get the 1000000th Fibonacci number, you need to divide the 1000001st Lucas number by √5. Your asnwer is true only if we assure that 0 is included in Fibonacci sequence (but the video doesn't include it).

[@diegorattaggi2095]

This is not correct. phi^1000000 is L_1000000 - phi^(-1000000) which is approximately L_1000000.

[@s0ngf0rx]

coding this up in python with matplotlib was so fun. thanks for this.

[@elliwesishawkins4799]

If you were to solve backwards for Fibonacci sequence, where as knowing that fn=f(n-1)+f(n-2) could be arranged so that determining, say, the numbers before the sequence officially starts and going backward would be finding f(n-2) as the number solved is behind the sequence. So the starting numbers of value 1, the 1 again we know the number before then would be 0, (so that 0+1=1) then before that would be 1, again before would be -1, then backwards to 2, backwards to -3, ect where it alternates between positive and negative numbers. Following this patterns backwards creates a mirror of the Fibonacci sequence where every other number is negative. And by “mirror” I mean reversely ordered from 0 as it comes from, presumably, a negative infinity to add its way down to 0, then back up to positive infinity in the recognized Fibonacci sequence.

[@martinepstein9826]

When you go backwards you get the Fibonacci sequence again but with alternating sign. So the ratio F(n+1)/F(n), where n is large and negative, approaches -0.618... i.e. the negative reciprocal of the golden ratio. This makes sense because 1.618... and -0.618... are the two eigenvalues of the matrix [1 1; 1 0] which is the matrix that generates the Fibonacci sequence.

[@elliwesishawkins4799]

Martin Epstein thank you, I was doing it myself and edited mine only to see yours and so mine now reflects yours. I appreciate you responding lol, I was about to do some crazy math myself.

[@elliwesishawkins4799]

Martin Epstein also my recognition that it was backwards and every other negative wouldn’t have given me the actual ratio, thank you very much

[@luciengrondin5802]

This concept of quasi certitude is interesting from an epistemological point of view. Never heard of it before, yet it's quite intuitive.

[@Johan323232]

A fun puzzle with Fibonacci numbers. If you take the Fibonacci recurrence and start with two positive numbers, it goes to infinity. If you start it with two negative numbers it goes to zero. However there are numbers you can start the sequence with and have it go to zero. Finding them is a fun way to practice computing recursion limits.

[@DeclanMBrennan]

Very interesting. If the terms are all one, this is essentially a random walk. By a vague argument of similarity, it seems like the length in this case should be [some constant] ^ N. However it turns out to be N ^ (0.5) .

[@DStecks]

It makes intuitive sense that the sequence will (almost) always grow because only very specific patterns will keep the values low, and if the values ever get larger, they compound. So only a tiny sliver of the possible results don't result in growth.

[@dansheppard2965]

I still wanna know about that pigeon photo!

[@arcanely]

What if the chance of a + was 2/3 and - was 1/3? What would the new growth rate be? Is there a way to generalize the growth rate for different probabilities?

[@briant6164]

Thanks!

[@yvesdelombaerde5909]

Even if you start de Fib. sequence with random numbers you get the Fn/Fn-1 -> phi 1.618… . So phi is the consequence of the cumulative summing process and not really related to the numbers in itself (0,1,1,2,3,5,8,…).

[@ayaipeeoiiu8151]

Is there a curve to know what’s the ratio for any randomnacci sequence (i mean for the regular fibonacci it’s (0;1)->1,6180339887... and for (0,5;0,5)->Viswanathan’s constant) and we can try to understand the pattern and maybe find a formula for these constants

[@kapa1611]

5:28 great name! reminds me of Ben Finegold xD it turns out that, if you continue this sequence a million times, you get Vishwanathin' :P

[@adolfadolph6504]

I THINK YOU WERE LOOKING FOR AN AGADMATOR VIDEO

[@georgettebeulah4427]

This makes so much sense

[@stormysamreen7062]

Am i the only one who sees a framed brown paper in the background?

[@YtseFrobozz]

I want to know what video that's from! Oh... never mind. It's the Graham's number video. OBVIOUSLY.

[@hojanson7331]

Correction The nth term of the Fibonacci sequence is φ^(n-1)

[@HunterJE]

Played around with this in a spreadsheet (yeah I know) and was fascinated how incredibly variable these can be in how quickly these blow up from one run to another...

[@klaasbil8459]

One could improve on the formula given at 1:58, by skipping the first part of the Fibonacci series where the ratios betweeen successive numbers are quite far from the Golden Ratio. As an example, the 20th Fibonacci number is 6764. So phi^1000000 could be approximated by 6765 * phi^999980. Skipping the first 1000 is most likely even better. Keep in mind that you need a lot of decimals in your phi value to get close to the correct answer!

[@halimk1777]

Very insightful video. I love everything related to the golden ratio and the fibonacci sequence, and especially when combined with randomness. I created an excel file which generates the finbonacci, the golden ratio, a random fibonacci sequence R(n+2) = R(n+1) + R(n) , the growth rates, and the Viswanath's number, which is the geometric average of all the growth rates. I also computed a very interesting number that has never been done before, (maybe It will carry my name someday :) which is the probability that the random fibonacci sequence breaks, that is when R(n) = 0. At this point the Viswanath's number cannot be computed as R(n+1)/R(n). And that has a probability higher than 50% that it occurs. Indeed of the first coin flip is a tails, the random fibonacci sequence breaks. I can send you the file if you want.

[@law5]

Hi! I would be really interested in this file! thx

[@thepseudoscientist1256]

I bet phi to the power 10000 would be pretty close to the 10000th LUCAS number

[@littleratblue]

I find it most interesting that the sequence forks between positive or negative and will follow a channel along one leg of the V. I would have been curious to see what the odds are that it would switch sides as n grows larger.

[@thezebraherd8275]

Do we know if the number is algebraic or transcendental?

[@MrYerak5]

When mathamaticions gets board they flip coins forever

[@moumous87]

I forgot that Fibonacci series had some relation to the Golden Ratio!

[@TimothyReeves]

moumous it's only mentioned in like every sqrt(5) Numberphile videos!

[@user-or5ke5yn4w]

Ron Graham's autograph at the wall:D

[@tux3506]

JAMES ❤️

[@krishnachattopadhyay3251]

There's no need of doing a toss there(3:38). The next term will be 1 in both the cases

[@ragnkja]

Abhirup Chattopadhyay No, it can be -1.

[@uditanshusadual7781]

Fibonacci ; I can do this all the day

[@johnathancorgan3994]

I can't decide if James or Holly has the most infectious enthusiasm for math.

[@ostrich_dog]

It would be interesting to see what the ratios are when we change the probability, say x% for getting + and 100-x% for getting -

[@soundnotestudios4896]

If you use this pattern (+,+,+,-,•) with the Fibonacci sequence, you come up with this number sequence: {1,1,2,3,5,-2,-10,-12,-22,-34,12,-408,-396,-804,-1200,396,-475200} and so on. I also experimented with other patterns such as (+,-,+,•) and (+,+,-) which ended up repeating itself. One conclusion I can come up with is that some patterns can continue on to positive or negative infinity unless -475200 can start back to 1. Side note: some patterns can also cancel out at 0 but forgot which pattern I used. I know this comment is out of context, but after I saw this video I thought to start coming up with these patterns to see what happens. Perhaps y’all should make a video on this. If anyone didn’t understand what I was explaining leave a comment and hopefully I could explain more.

[@joshualowe6950]

2:28 that’s the 1000001th number though. To find the 1st term phi^0 as you take 1 and do not multiple by phi to get it. To get the 2nd term, 1xphi^1 (approx obviously), 3rd term 1xphi^2 so 1000000th term is 1xphi^999999. Fn ~= phi^(n-1)

[@AbhisekBagishere]

Fascinating

[@FanTazTiCxD]

If the lead singer from Radiohead Thom Yorke wasn't in Radiohead, but was a mathematician instead