Discrete Math. Instructor: Mike Picollelli. Day 10

Transcription

1 Day 10

2 Fibonacci Redux. Last time, we saw that F n = 1 5 (( 1 + ) n ( ) n ) 5. 2

3 What Makes The Fibonacci Numbers So Special? The Fibonacci numbers are a particular type of recurrence relation, a second-order linear recurrence relation with constant coefficients: F n = F n 1 + F n 2.

4 What Makes The Fibonacci Numbers So Special? The Fibonacci numbers are a particular type of recurrence relation, a second-order linear recurrence relation with constant coefficients: F n = F n 1 + F n 2. Second-order means only the previous two values determine the next (i.e. F n depends on F n 1 and F n 2 but not F n k for k > 2). Linear means we re not multiplying the numbers together in our relation. Constant coefficients means the coefficients of F n, F n 1, F n 2 don t depend on n at all.

5 So What Do We Do In General? Suppose we re given a linear recurrence relation with constant coefficients that s of order k, so that we know T 0, T 1,..., T k 1, and, for n k we have T n = C 1 T n 1 + C 2 T n C k T n k.

6 So What Do We Do In General? Suppose we re given a linear recurrence relation with constant coefficients that s of order k, so that we know T 0, T 1,..., T k 1, and, for n k we have T n = C 1 T n 1 + C 2 T n C k T n k. First, we find the characteristic polynomial. That s just the polynomial t k C 1 t k 1 C 2 t k 2 C k 1 t C k.

7 So What Do We Do In General? Then, through whatever magic is necessary, we find the roots of the characteristic polynomial. According to the Fundamental Theorem of Algebra, there are k (possibly complex and including repetition) roots.

8 So What Do We Do In General? Then, through whatever magic is necessary, we find the roots of the characteristic polynomial. According to the Fundamental Theorem of Algebra, there are k (possibly complex and including repetition) roots. If the roots are all distinct (none is repeated), say t 1, t 2,..., t k, then k T n = A i (t i ) n, i=1 where the A i are constants that we compute using the values of T 0,..., T k 1. Just like before.

9 Some Other Important Numbers. Definition: Let a, b N. Then we say a divides b, written a b, if b = ka for some k N. We call a a divisor of b.

10 Some Other Important Numbers. Definition: Let a, b N. Then we say a divides b, written a b, if b = ka for some k N. We call a a divisor of b. Definition: A number p N, p > 1, is prime if its only divisors are 1 and p.

11 Some Other Important Numbers. Definition: Let a, b N. Then we say a divides b, written a b, if b = ka for some k N. We call a a divisor of b. Definition: A number p N, p > 1, is prime if its only divisors are 1 and p. A number n N, n > 1, is composite if it is not prime.

12 Some Other Important Numbers. Definition: Let a, b N. Then we say a divides b, written a b, if b = ka for some k N. We call a a divisor of b. Definition: A number p N, p > 1, is prime if its only divisors are 1 and p. A number n N, n > 1, is composite if it is not prime. 1 is neither prime nor composite.

13 Some Properties Of Primes. Whenever p ab for a, b N, we must have p a or p b.

14 Some Properties Of Primes. Whenever p ab for a, b N, we must have p a or p b. Every n N, n > 1, is the product of primes.

15 Some Properties Of Primes. Whenever p ab for a, b N, we must have p a or p b. Every n N, n > 1, is the product of primes. The Fundamental Theorem of Arithmetic: Every n N, n > 1, can be written uniquely as a product of primes, i.e. n = p e 1 1 pe 2 2 pe k k, where p 1 < p 2 < < p k are primes, and the exponents e 1,..., e k are uniquely determined by n.