## MA-410 Homework 1 Due at 4:59pm in my mailbox in SAS 3151, Tuesday, February 5, 2019

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 n-th Chebyshev-3 Polynomial Vn(x) in the variable x with integer coefficients, defined by the linear recurrence V0(x) = 1, V1(x) = 2x-1, Vn+2(x) = 2xVn+1(x) - Vn(x) for all n ≥ 0 one has
(y+1/y) Vn( (y2 + 1/y2)/2 ) = y2n+1 + 1/y2n+1 for all n ≥ 0, real numbers y ≠ 0.
2. Consider Cn from ENT, §1.2, Problem 10, page 12. Prove that Cn = binomial(2n,n) - binomial(2n,n-1) for all n ≥ 1, thus showing that the Cn are integers.
3. ENT, §2.2, Problem 20(f), page 25. Prove first that GCD(a,b2) = 1, and then GCD(A^2, B) = 1 for A=a and B = b^2.
4. Please compute integers x,y,z such that 20x + 28y + 35z = 1. Please show your work, computing a solution to 20x + 28y = 4 and 4w + 35z = 1 first.
5. Prove for the extended Euclidean algorithm described in class: sk tk+1 - sk+1 tk = (-1)k+1 for all -1 ≤ k ≤ n