Presentation on theme: "Fibonacci Numbers and Binet Formula (An Introduction to Number Theory)"— Presentation transcript:
1Fibonacci Numbers and Binet Formula (An Introduction to Number Theory) By: (The Ladies)2
2Recurrence Sequenceeach 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
3Mathematical induction Fibonacci (1,1,2,3,5,8,13...)1,1+1,1+1+2, , ,1,2,4,7,12,20...+1 to each term2,3,5,8,13,21...because(2+1)(3+2)+3+5+8(5+3)+5+8(8+5)+8(13+8)
4Fibonacci sequence patterns neither arithmetic nor geometricso write it in a different way1/1=12/1=(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)
5Golden Ratio φφ=1+1/φφ2=φ+1 quadratic equationφ=(1+sqrt(5))/2 (only the positive answer)φ=
6Golden Ratio and practical application most famous and controversial in history - human aestheticsConverting between km and miles1 mile= km13 km = 8 milesFibonacci (1,1,2,3,5,8,13,21...)OK, using Fibonacci numbers, how many miles are in 50 kilometers?? (show your work)
7Binet FormulaA formula to find a term in a Fibonacci numbers without generating previous termsJacques Binet in known to Euler and Bernoulli 100 years beforeFibonacci numbers are actually a combo of two geometric progressionsRecall φ2=φ+1 and τ2=τ+1 identitiesUse them to come up with a formula for the Fibonacci series
8Binet Formula φ2=φ+1 and τ2=τ+1 identities φ2= φ+1 φ3=φ(φ2)=φ(φ+1)=(φ2)+φ=(φ+1)+φ= 2φ+1φ4=φ(φ3)=φ(2φ+1)=(2φ2)+φ=2(φ+1)+φ= 3φ+2φ5=φ(φ4)=φ(3φ+2)=(3φ2)+2φ=3(φ+1)+2φ= 5φ+3φ6=φ(φ5)=φ(5φ+3)=(5φ2)+3φ=5(φ+1)+3φ= 8φ+5φ2=1φ So, φn=Fnφ+Fn-1φ3=2φ andφ4=3φ τn=Fnτ+Fn-1φ5=5φ+3φ6=8φ+5
9Binet 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))/2therefore (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 isF1000= (φ^(10000) - τ^(10000)) / sqrt(5) = (209 digits)
14hn = # 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.
15h3 = 7 h5 = 31 h7 = 123 h4 = 15 h6 = 63 h8 = 247 What We Know: hn = small disks + big disk + small diskshn hn-12hnRecursive Formula: hn= 2hn-1+ 1
16Closed Formula:3, 7, 15, 31, 63, ...One less than a power of 2?3 =7 =15 =hn = 2n -1
17Prediction : 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?
18Prediction : Solution Number of moves required 264 -1 So . . . roughly 583,344,214,028 years.
19Prime Numbers: How do we find them? 200 B.C. Eratosthenes invented the sieve.
25Why? The number of tests is the # of primes < testing maximum Proof by contradiction:1. A composite exists in2. Thus, it is not a multiple of a P < 103. Thus, both factors > 104. Therefore, the composite > 100
26Prime Numbers: How many exist? E = P1 * P2 * P3 * P4 ... Pnnow...q = P1 * P2 * P3 * P4 * ... * Pn + 1Following the Composite Theorems (must be factor of unique prime numbers), infinite prime numbers exist.
27Where Aren't the Prime Numbers? 2*3 + 2 = composite 2*3 + 3 = compositeK = 2 * 3 * 4 * ... * (N+1)K+2 = 2 * 3 * 4 * ... * (N+1) +2K+3 = 2 * 3 * 4 * ... * (N+1) + 3K+(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.