## MA-410 Spring 2024 Homework 1 Due as indicated for each problem.

All solutions must be submitted on the Moodle web site for the class at wolfware.ncsu.edu. You may upload a photo of your handwritten solution or a file of your typed solution.
Note my office hours on my schedule.

1. Due Thurs January 25, 11:59pm.
Please prove by induction that for the n-th Chebyshev Polynomial of the First Kind Tn(x) in the variable x with integer coefficients [https://en.wikipedia.org/wiki/Chebyshev_polynomials], defined by the linear recurrence T0(x) = 1, T1(x) = x, Tn+2(x) = 2xTn+1(x) - Tn(x) for all n ≥ 0 one has
Tn( (y + 1/y)/2 ) = (yn + 1/yn)/2 for all n ≥ 0, real numbers y ≠ 0.
2. Due Thurs January 25, 11:59pm.
Consider the Catalan numbers as defined Cn = binomial(2n,n) / (n+1) in ENT, §1.2, Problem 10, page 12 for all n ≥ 0. Prove that Cn = binomial(2n,n) – binomial(2n,n–1) for all n ≥ 1, thus showing that the Cn are integers.
3. Due Tues January 30, 11:59pm.
ENT, §2.2, Problem 20(f), page 25: If gcd(a, b) = 1, then gcd(a2, b2) = 1.
Hint: prove first that GCD(a,b2) = 1, and then GCD(A^2, B) = 1 for A=a and B = b^2; or use the Fundamental Theorem of Arithmetic.
4. Due Tues January 30, 11:59pm.
Please compute integers x,y,z such that 90x + 99y + 110z = 1. Please show your work, computing a solution to 90x + 99y = 9 and 9w + 110z = 1 first. Note: you only need to give one triple.
5. Due Tues February 6, 11:59pm.
Prove for the extended Euclidean algorithm described in class: sk tk+1 – sk+1 tk = (–1)k+1 for all –1 ≤ k ≤ n–1.