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, , ,
1,2,4,7,12,20...
+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)

4. 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)

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

6. 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)

7. 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

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

9. 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)

10. The Fibonacci Sequence in NatureX0

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. 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.

15. 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

16. Closed Formula:
3, 7, 15, 31, 63, ...
One less than a power of 2?
3 =
7 =
15 =
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
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.