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

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

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

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

Fibonacci sequence patterns
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)

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

Golden Ratio and practical application
most famous and controversial in history - human aesthetics Converting between km and miles 1 mile= 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)

Binet Formula A formula to find a term in a Fibonacci numbers without generating previous terms Jacques Binet in 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

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 So, φn=Fnφ+Fn-1 φ3=2φ and φ4=3φ τn=Fnτ+Fn-1 φ5=5φ+3 φ6=8φ+5

Binet Formula Fn= (φ^n - τ^n) / (φ - τ) φ^n=Fnφ+Fn-1 τ^n=Fnτ+Fn-1
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) = (209 digits)

The Fibonacci Sequence in Nature
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Application : The Towers of Hanoi

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.

An Example: 1 2 3 4 5 6 7

hn = # of moves required to transfer n disks.
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.

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

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

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?

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

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

Prime Numbers:

Prime Numbers:

Prime Numbers:

Prime Numbers:

Prime Numbers: And it stops.

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

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.

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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