lucas

  1. D

    Help with understanding the proof of Lucas' formula (strong version of induction)

    Hi. I don't understand the proof of Lucas' formula. I would be very happy is someone could explain it step by step. Especially the last to lines of the proof. Here is a link to the proof. - Gøran
  2. M

    Generalized Fibonacci and Lucas Numbers.

    Can you help me prove this theorem regarding Fibonacci and Lucas numbers? Theorem. Let m,r ϵ Z and n be non-zero integer. Then U2mn+r ≡ (-1)mn Ur (mod Um) and V2mn+r ≡ (-1)mn Vr (mod Um). Im not that good at proving. This type of congruence is much harder than what I read in our book...
  3. E

    Lucas Sequences

    I read some facts about Lucas sequences, but they were not proven, can you help me? Let P and Q be integers. Define u_{n+1} = Pu_n - Qu_{n-1} with u_0 = 0, u_1 = 1. Similar define v_{n+1} = Pv_n - Qv_{n-1} with v_0 = 2, v_1 = P. Show: (1) 2u_n = Pu_{n-1} + v_{n-1} (Managed to prove by...
  4. A

    Proof about Fibonacci and Lucas numbers (GCD)

    Hi. How to prove that GCD(F3k, L3k)=2? and If n dosen't divide with 3 GCD(Fn, Ln)=1 Don't even know how to start that:/....Hope someone can help:) Ty.
  5. Deadstar

    Ironing out the Lucas Primality Test proof

    As I was typing this out I realized I had more questions than I thought... Theorem Let n > 1, and suppose that for every prime factor q of n-1 there is an integer a such that, a^{n-1} \equiv 1 (\textrm{mod } n) but, a^{\tfrac{n-1}{q}} \not\equiv 1 (\textrm{mod } n) Then n is prime. Proof...
  6. D

    help with Lucas numbers

    Given L_{1}=1,L_{2}=3,L_{n+2}=L_{n}+L_{n+1}, how can I prove that L_{3}=\alpha^3-\alpha^{-3} where \alpha=\frac{1+\sqrt{5}}{2}??
  7. Z

    Lucas Numbers

    Not sure if I'm putting this in the right forum, but its a high school question. Why is the first Lucas Number 2? and the second 1? Thanks
  8. K

    Lucas Number Proof By Induction

    I am having some trouble proving the relationship F_n L_n=F_{2n} where F_n, L_n are the Fibronachi and Lucas numbers respectively. Oh, and how should I have written this in Laytex?
  9. H

    Proving Lucas Numbers and Fibonacci Numbers

    The Lucas numbers l_{0}, l_{1}, l_{2},...,l_{n}... are defined on the same recurrence relation defining the Fibonacci numbers, but the Lucas numbers posses different initial conditions. l_{n}=l_{n-1} + l_{n-2}, (n≥2), l_{0}=2, l_{1} = 1 (a)l_{n} = f_{n-1} + f_{n+1} for n≥1. My hardest part...
  10. S

    Fibbonacci and Lucas Number

    Ok. I have made a list of the first 30 lucas and fibbonacci number, and the products they make. I now need to find a relationship between the products and the fibonacci numbers. I know thats a bit vague but that is what the teacher told us to do, and he said it was blatantly obvious, so either...
  11. M

    Lucas Sequence

    Here is my problem, two parts. The Lucas sequence is defined by (L_n) = {1,3,4,7,11,18,...} where the successive terms is found by adding the two previous terms. 1.) Show that the general term, or the nth term of the Lucas sequence is L_n = [(1 + SQRT(5))/2]^n + [(1 -...
  12. Y

    Prove L_n = L_n-1 + L_n-2 for n >= 3, where L_n are Lucas numbers.

    Prove L_n = L_n-1 + L_n-2 for n >= 3, where L_n are Lucas numbers. Thank you!