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Lucas Numbers – Numberphile

Lucas Numbers – Numberphile


MATT PARKER: Okay, so, Lucas Numbers are named after a guy called Edward Lucas Although, he was French, so ‘Loo-cah’, but I’m going to say ‘Lu-cas’ because I am lazy And if it wasn’t for Lucas, we wouldn’t really care about Fibonacci, because Fibonacci did some things he wrote those numbers down, didn’t generalize them, didn’t analyze them He did introduce some interesting number systems to Europe, and he did some other cool bits and pieces but these are more of a footnote to the story of his life and Lucas went, “Hey, these are really
interesting”, and he’s the guy who promoted them and he went with his own sequence which is better than the Fibbonacci Numbers Then you think, okay, if Fibbonacci starts 1,1 what is our next option? So Fibonacci starts 1, 1 and then it goes 2 and then 3 and then 5 and then 8 and then 13 and then onwards Well, what if we start — instead of starting 1, 1 what if we start 1, 2? but we can’t start 1, 2 because we’re just starting here All we’ve done is move along slightly and so we’re going to get the same numbers if we start 1, 2 Lucas numbers, you start 2, 1 — so you just swap them around and then you repeat exactly the same pattern So 2 + 1 then gives us 3 and then we get 4 and then you get 7 and then you get 11 and 18 And these are the Lucas numbers! And to my mind this sequence of numbers is far more interesting then the Fibonacci sequence of numbers, and these have a stronger link to the golden ratio So if you take the golden ratio Φ (phi), which is equal to 1.61803398… and then a 9, and so on Okay, right, so we know for any of these sequences they’re going to approach that ratio But what if we start with the ratio, what if we want to take the ratio and then get a sequence of numbers? So if you calculate Φ² — Φ²=2.6180…, which if you round to the nearest whole number is 3 (approximately) Okay, so I’m going to do this column as rounding to the nearest whole number if you do Φ³ — Φ³=4.23606…, which if you round that to the nearest whole number is 4 Φ⁴=6.85410…, and rounded to the nearest whole number is 7 Φ⁵ — and if I carry on and take each of these and round them to the the nearest whole number, what do I get? Sure enough, these are the Lucas numbers appearing up here, and that will carry on all the way down Every power of the golden ratio beyond the nearest whole number gives you the Lucas numbers and so that’s not the fact that the Fibonacci numbers are linked to the golden ratio in a way that every other similar sequence is. If you start with the golden ratio and then take powers — is because what you’re doing is multiplying each term by the golden ratio to get the next term which is that wonderful property that everyone bangs on about But these are the numbers it generates, if you start with that and then go to the sequence and that’s why I think the Lucas numbers are vastly superior I’ve been very careful, I’ve never said Lucas sequence, which is a very good point, because Lucas sequence is, well, a family of other things. So Lucas did some amazing research into all the weird properties you get from these various sequences and these numbers were just one specific example of a Lucas sequence but they’re the most common one and the most famous one, which is why they are called the Lucas numbers but there are all sorts of amazing things you can do these numbers You can — everything from obviously wonderful ratios but you can use them to test if a number is prime and, I mean, there is a whole amazing area of math Lookup Lucas sequences, there is some incredible stuff in there BRADY: One thing about this that strikes me though is we often talk about how mathematics is very precise and perfect and this great thing This sequence does match, but it only matches by kind of roughly rounding, and it seems like each time it kind of slightly misses the target but it misses by such a small amount that it doesn’t matter This doesn’t seem very much like a lot
of the other mathematics I see it seems kind of a bit fuzzy and “almosty” DR. PARKER: I think rounding — people rain hate on rounding because you do it at primary school and people view it as maths you did very early on, and very simple and there’s kind of a false — People link mathematics with unnecessary precision and they link mathematics with doing things that are pointless and over the top and so, for some reason, one of the first things people would do when they’re learning math at high school and want to look smart is write down an unnecessary number of decimal places Right? because they think it’s more maths-y — the more numbers you write down, the more maths we’re doing and I think it’s a shame, there are some amazing things that drop out of rounding and so I think it’s nice that the Lucas numbers you do get these wonderfully precise answers out of the powers then you round to the nearest whole number and that’s where the beauty is, I like that.
]MATT PARKER: 8,229 + 17,399=45,628 so I’ll do it for a few of these and you can zip through it obviously BRADY: So is this the Brady sequence? This is the Brady sequence! We’re going to call these the Brady numbers.

100 thoughts on “Lucas Numbers – Numberphile”

  1. I'd like to know: if instead of taking phi, phi^2, phi^3 and so on and round and get the Lucas numbers, you take a constant c and then consider c phi, c phi^2, c phi^3 and so on and then round. Depending on a, you get different sequences. Can you find and a such that the sequence is the Fibonacci sequence? If yes, what is that number?

  2. my immediate thought is there any relationship between "left overs" after rounding? Some pattern to phi^n-round(phi^n) ? the "round" being a pseudo computer program command to round the number, not sure how that should be written in math speak 🙂

  3. Isn't phi to the first power 1.61…closer to 2? But that doesn't fit the Lucas numbers, did I miss something where he said that the first power and lower wouldn't count or something.

  4. Side comment: how funny to see that the visual mark of the channel is the type of paper used. Rather unique I have to say.

  5. Honestly I find rounding very useful
    I made a formula for finding a phytagoras triple using rounding and other techniques usually considered useless

  6. I'm going to invent my own sequence so I can be known for something. How about you start with 1. And then you repeat the number 1. So start with 1 and then the second number is 1 and the third number is another 1 and so on and you just keep repeating that for infinity. What do I win?

  7. 1. That is weird with the 2 then 1 thing at the start.

    2. I was into unnecessay precision. I did an entire app to endulge that.

  8. The Fibonacci numbers have something else to do with the Golden Ratio.
    1 * phi is about 2
    2 * phi is about 3
    3 * phi is about 5
    5 * phi is about 8

  9. Actually for as long as Excel's precision goes, Fibonacci sequence, starting with 1,2,3, is always one step ahead than Lucas' numbers… (That is to say that 5/3, 8/5, 13/8 and so on is closer to φ than 4/3, 7/4, 11/7 and so on)

    EDIT: Even if we start properly, with 1,1,2,3.. Fibonacci is still better!

  10. You seriously not gonna point out that Phi^n+Phi^(n+1)=Phi^(n+2) for all the numbers you listed? Even while we're rounding?

  11. in fact if you use the golden ratio again to write phi^n you can find Fibonacci again. Nothing spectacular about Lucas Numbers:
    phi^1= phi + 0
    phi^2= phi + 1
    phi^3= 2phi + 1
    phi^4= 3phi + 2
    phi^5= 5phi + 3

  12. Actually, I have found some rather interesting connection between the Fibonacci numbers and Phi. I am too lazy to write out how I got to this, so I'll just write the general form:

    (Phi^n = Fn*Phi + F(n-1
    For any positive integer n, where Fn represents the nth Fibonacci number, and F(n-1) represents the Fibonacci number before the nth.

    And, well, I mean, someone most likelyfound that a long ago, but whatever.

  13. wow… the property of Lucas number actually came in my discrete math course before which is asked to prove from the general formula of it…

  14. Love your channel! It seems to me however that these numbers are not "superior" because of the power property, as the power property of the Lucas numbers is a necessary consequence of the fact that the quotient between two subsequent numbers in any sequence in the "Fibonacci family" (don't what is the real name) converges to the golden ratio as n goes to infinity. You can see it already in the video, where the errors are decreasing as n increases. Therefore, the nth term of a Fibonacci sequence with any starting terms can eventually be written pretty accurately as c*phi^n (where phi is the Golden Ratio and provided n is big enough). In case of the classic Fibonacci sequence, this c is equal to about 0.7243606… (according to my excel sheet)! It's just convenient beauty that the Lucas numbers have a c equal to 1.

    That is to say – they are both equally beautiful! No discrimination in mathematics madafakkaaa. Unless you're solving quadratic equations of course.

  15. Well, actually, Fibonacci also have a nice form of calculating the closed forms (using linear recurrence), but I suppose you know that.

  16. I don't like decimal places if they are infinitely long. In those cases, give me representations by functions or give me death!

  17. lucas = zeros (100,1);
    lucas (1) = 2;
    lucas (2) = 1;
    for i=3:100
    lucas (i) = lucas (i-1) + lucas (i-2);
    end

    phi = (1+sqrt(5))/2;

    t = zeros (100,1);

    for i=1:100
    t(i)=round(phi^(i-1));
    end

    v==t

    I executed this MatLab code and the result is that from a point the rounding terms are not same as the Lucas numbers( except for the first two terms),Is that from the MatLab error in rounding on decimals?
    Is there any proof that rounding powers of phi gets you Lucas Numbers?

  18. I know this is kinda old, but as you raise Phi to higher powers, it gets extremely close to whole numbers. like by the time you raise it to the 25th power or something it's about within 0.000003 of a whole number

  19. By extension, if you start the sequence 1, Φ… you get the exact powers of Φ:
    1, Φ, Φ^2, Φ^3, Φ^4
    And this goes backward, if you start with any two consecutive powers of Φ in order (e.g. Φ^-10, Φ^-9) it will generate all the powers exactly.

  20. 0:18 – it just occured to me: Fibonacci was really named Bonacci, so his nickname is Fi-Bonaccci, you can spell it Phi-Bonacci or even Φ-Bonacci. He has Golden Ratio in his name!

  21. but fibonacci starts at 0 and has no special starting conditions.. so for me it looks way more natural. make a spiral form the other numbers.. ye u cant.. so wtf. it just approaches the golden ratio faster..
    while fibonacci numbers can be easily shown geometrically
    sry my engrish is not very london..

  22. phi^0 is 1 and phi^1 is 2 and then 3, 4, 7 and so on but lucas numbers aren't 1 2 3 4 7 11… so…?

  23. So the Lucas numbers are like the Parker square?

    Wait… what happens if you round the square? Will it equal 4?

  24. Those Lucas Numbers are wild! What kind of coffee are they drinking? Fun video that stimulates my curiosity very very deeply!

  25. "…people link mathematics with doing thing that are unnecessary and over the top…"

    What if that's why I like mathematics?

  26. I have discovered a cool explanation of why the Lucas series begins 2,1 , it will sound slightly strange to begin but I'm sure it makes sense))). The 2 means that in this series each number is the sum of the 2 previous numbers, And the 1 is 1 less than 2 to the power of 1. I have discovered some other Ratios associated with series' where each number is the sum of its 3,4,5 or 6 predecessors, when we take the powers of these ratios which extend towards 2, we have an equivalent Lucas series for each ratio. So the powers of the third ratio (1.839286755) round towards the 3,1,3 series, the 3 means each number is the sum of its 3 predecessors and the 1 and 3 are 1 less than 2 to the 1 and 2 squared. The 4th Ratio rounds to the 4,1,3,7, series then 5,1,3,7,15 and 6,1,3,7,15,31.My spreadsheet wont allow me any further so my progress has slowed somewhat, I'm not sure if the tenth powers rounds to the series beginning 1 or 10. This is another reason why the Lucas series is King as it ties these other ratios together with phi.

  27. Well, if you want Xn = φ^n, then choose X0 = 1, and X1 = φ. Since φ² = φ + 1, then φ^n  = φ^(n-1) + φ^(n-2), and this way we indeed have Xn = X(n-1) + X(n-2), and Xn = φ^n.

  28. There is no rounding needed.
    2 = Phi^0 + Phi^0
    1 = Phi^1 – Phi^-1
    3 = Phi^2 + Phi^-2
    4 = Phi^3 – Phi^-3
    7 = Phi^4 + Phi^-4

  29. if you measure the area of a swimming pool
    the result could be obtained using centimeters or millimiters, any more precision is not practically usable by humans and the result would not be more meaningful

  30. Anyway, we round to get phi in Fibonacci too. 1/1 is 1, 2/1 is 2, 3/2 is 1.5, 5/3 is 1.666… (and it starts to look closer to phi), and so on. Only with n that tends to infinity we can say that Fn/Fn-1 = phi…

  31. The last digit of the Lucas Numbers always repeat themselves in a sequence! 4, 7, 1, 8, 9 7, 6, 3, 9, 2, 1, 3, … and now if you add two adjacent numbers n_m and n_{m + 1} then the last digit of the result will be n_{m + 2} so it is a cycle!

  32. Why don't you go over that L_n (the nth Lucas number) is equal to (phi)^n + (phi_hat)^n where phi_hat is (1-sqrt(5))/2?

  33. I always round to two decimal places it feels right. Not so imprecise to lose its value, but not so precise that it takes up too much space. I never round on certain in-between steps, like converting grams to moles, I round at the end. It’s like if π was rounded to 3.1, diameters greater than 5 units in length would lose precision, but 3.141592 is just unnecessary.

  34. Technically, besides a sequence 0, 0, 0, … aren't all other such sequences better than Fibonacci? That is, isn't the Fibonacci sequence the one that converges the most slowly to phi?

  35. Watching this 4 years later and that's still an incredibly dodgy, hand-wavey answer to the problem with rounding. It was basically "I like it". The rounding, however, still remains imprecise and arbitrary.

  36. Has anyone calculated the value of the exponents needed to modify phi to make it equal to each of the Lucas numbers to see if there's an interesting pattern that emerges as a result?

  37. There is a little mistake in the video, as it shouldn't say that Lucas numbers are an approximation to ( ( 1 + SQRT( 5 ) ) / 2 )^n. Instead, it should have said that there is an equivalent way to build them which is, exactly:

    Ln = ( ( 1 + SQRT( 5 ) ) / 2 )^n + ( ( 1 – SQRT( 5 ) ) / 2 )^n.

    The second term, by the way, is the second solution to the golden equation: phi + 1 = phi ^ 2. I guess that solves the discussion about the unnecessary rounding.

  38. Actually, the only reason the Lucas numbers are so close to powers of φ is because 2 is the closest integer to φ.
    If you started the sequence with, say, 1.6 and 1, the values would be a lot closer, and if you started with φ and 1, the values would be exactly equal.

  39. We needn't make do with an "almosty" link between the Lucas numbers and phi: the nth Lucas number is exactly
    phi^n+(-1)^nphi^{-n}
    Rounding is equivalent to ignoring the latter term.

  40. Since it's not super precise and doesn't quite work. Does this mean that Lucas numbers are a kind of Parker sequence?

  41. am i the only one to realize something? Take a term from the fibonacci sequence and add it with the second previous term instead of the previous. If you do that with the whole series, you get a familiar sequence c;

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