Day 10
Fibonacci Redux. Last time, we saw that F n = 1 5 (( 1 + ) n ( 5 2 1 ) n ) 5. 2
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.
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.
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 2 + + C k T n k.
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 2 + + 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.
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.
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.
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.
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.
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.
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.
Some Properties Of Primes. Whenever p ab for a, b N, we must have p a or p b.
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.
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.