MA-410 Homework 1

Due at 4:59pm in my mailbox in SAS 3151, Thursday, February 2, 2017

All solutions must be submitted in hardcopy either to me in class or placed in my mailbox.
Note my office hours on my schedule.

1. Prove by induction that for the rational function F_n(x) in the variable x, defined by the quadratic recurrence F_0 = x+(1/x), F_{n+1}(x) = F_n(x)^2 - 2 for all n ≥ 0 one has
   
   F_n(x) = x^(2^n) + x^(-2^n).
   
2. Consider C_n from ENT, §1.2, Problem 10, page 12. Prove that C_n = binomial(2n,n) - binomial(2n,n-1) for all n ≥ 1, thus showing that the C_n are integers.
3. ENT, §2.2, Problem 20(e), page 25.
4. Please compute integers x,y,z such that 35x + 45y + 63z = 1.
5. Prove for the extended Euclidean algorithm described in class: s_k t_k+1 - s_k+1 t_k = (-1)^k+1 for all -1 ≤ k ≤ n