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Fibonacci Numbers and Binet Formula (An Introduction to Number Theory)

Fibonacci Numbers and Binet Formula (An Introduction to Number Theory). By: ( The Ladies ) 2. Recurrence Sequence. each further term of the sequence is defined as a function of the preceding terms (starting seed and rule) Fibonacci sequence (1,1,2,3,5,8,13,21,34,55...)

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Fibonacci Numbers and Binet Formula (An Introduction to Number Theory)

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  1. Fibonacci Numbers and Binet Formula (An Introduction to Number Theory) By: (The Ladies) 2

  2. Recurrence Sequence • each further term of the sequence is defined as a function of the preceding terms (starting seed and rule) • Fibonacci sequence (1,1,2,3,5,8,13,21,34,55...) • Lucas sequence (2,1,3,4,7,11,18,29,47,76) • take 3+8 (1+2)+(3+5) (1+3)+(2+5) (4)+(7) - can be shown to hold in general

  3. Mathematical induction • Fibonacci (1,1,2,3,5,8,13...) • 1,1+1,1+1+2,1+1+2+3,1+1+2+3+5,1+1+2+3+5+8... • 1,2,4,7,12,20... • +1 to each term • 2,3,5,8,13,21... • because 1+1+2+3+5+8 1+1+1+2+3+5+8 (2+1)+2+3+5+8 (3+2)+3+5+8 (5+3)+5+8 (8+5)+8 (13+8)

  4. Fibonacci sequence patterns Fibonacci (1,1,2,3,5,8,13...) neither arithmetic nor geometric so write it in a different way 1/1=1 2/1=(1+1)/1 = (1+(1/1)) 3/2=(2+1)/2=1+(1/2)= 1+ 1/(1+(1/1)) 5/3=(3+2)/3=1+(2/3)=1+ 1/(1+ 1/(1+(1/1))) and so on (Golden ratio φ)

  5. Golden Ratio φ φ=1+1/φ φ2=φ+1 quadratic equation φ=(1+sqrt(5))/2 (only the positive answer) φ=1.618033989...

  6. Golden Ratio and practical application • most famous and controversial in history - human aesthetics • Converting between km and miles • 1 mile= 1.6093 km • 13 km = 8 miles • Fibonacci (1,1,2,3,5,8,13,21...) • OK, using Fibonacci numbers, how many miles are in 50 kilometers?? (show your work)

  7. Binet Formula • A formula to find a term in a Fibonacci numbers without generating previous terms • Jacques Binet in 1843 - known to Euler and Bernoulli 100 years before • Fibonacci numbers are actually a combo of two geometric progressions • Recall φ2=φ+1 and τ2=τ+1 identities • Use them to come up with a formula for the Fibonacci series

  8. Binet Formula • φ2=φ+1 and τ2=τ+1 identities • φ2= φ+1 • φ3=φ(φ2)=φ(φ+1)=(φ2)+φ=(φ+1)+φ= 2φ+1 • φ4=φ(φ3)=φ(2φ+1)=(2φ2)+φ=2(φ+1)+φ= 3φ+2 • φ5=φ(φ4)=φ(3φ+2)=(3φ2)+2φ=3(φ+1)+2φ= 5φ+3 • φ6=φ(φ5)=φ(5φ+3)=(5φ2)+3φ=5(φ+1)+3φ= 8φ+5 • φ2=1φ+1 So, φn=Fnφ+Fn-1 • φ3=2φ+1 and • φ4=3φ+2 τn=Fnτ+Fn-1 • φ5=5φ+3 • φ6=8φ+5

  9. Binet Formula φ^n=Fnφ+Fn-1 τ^n=Fnτ+Fn-1 φ^n - τ^n=Fnφ - Fnτ Fn= (φ^n - τ^n) / (φ - τ) remember that φ = (1+sqrt(5))/2 and τ = (1+sqrt(5))/2 therefore (φ - τ) = sqrt (5) Fn= (φ^n - τ^n) / sqrt(5) Fn=(φ^n/sqrt(5)) - (τ^n/sqrt(5)) (two geometric progressions) now for the Fibonacci term 1000 is F1000= (φ^(10000) - τ^(10000)) / sqrt(5) =43466557686937456... (209 digits)

  10. The Fibonacci Sequence in Nature http://www.youtube.com/watch?v=ahXIMUkSXX0

  11. Application : The Towers of Hanoi

  12. The Rules: From here:To here: Without: 1. Moving more than one disk at a time. 2. Placing a larger disk on top of a smaller disk.

  13. An Example: 1 2 3 4 5 6 7

  14. The Goal: To find the minimum number of moves necessary to complete the puzzle. hn = # of moves required to transfer n disks. Let us find a recurrence rule to predict hn.

  15. What We Know: h3 = 7 h5 = 31 h7 = 123 h4 = 15 h6 = 63 h8= 247 hn = small disks + big disk + small disks hn-1 + 1 + hn-1 2hn-1 + 1 Recursive Formula: hn= 2hn-1+ 1

  16. Closed Formula: 3, 7, 15, 31, 63, ... One less than a power of 2? 3 = 22 - 1 7 = 23 - 1 15 = 24- 1 hn= 2n-1

  17. Prediction : End of the World? High on the mountaintops sat a monk who could foretell the end of the world. He had a Tower of Hanoi with 64 gleaming diamond disks and could move one a second. When he stopped the world would end. How long do we have?

  18. Prediction : Solution Number of moves required 264 -1 So . . . roughly 583,344,214,028 years.

  19. Prime Numbers: How do we find them? 200 B.C. Eratosthenes invented the sieve.

  20. Prime Numbers:

  21. Prime Numbers:

  22. Prime Numbers:

  23. Prime Numbers:

  24. Prime Numbers: And it stops.

  25. Why? The number of tests is the # of primes < testing maximum Proof by contradiction: 1. A composite exists in 11-100 2. Thus, it is not a multiple of a P < 10 3. Thus, both factors > 10 4. Therefore, the composite > 100

  26. Prime Numbers: How many exist? E = P1 * P2 * P3 * P4 ... Pn now... q = P1 * P2 * P3 * P4 * ... * Pn + 1 Following the Composite Theorems (must be factor of unique prime numbers), infinite prime numbers exist.

  27. Where Aren't the Prime Numbers? 2*3 + 2 = composite 2*3 + 3 = composite K = 2 * 3 * 4 * ... * (N+1) K+2 = 2 * 3 * 4 * ... * (N+1) +2 K+3 = 2 * 3 * 4 * ... * (N+1) + 3 K+(N+1) = 2 * 3 * 4 * ... * (N+1) + (N+1) K+2, K+3, K+4, K+(N+1) --> all composite, there are runs infinitely long where there are no primes.

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