MA410 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.

Due Thurs January 25, 11:59pm.
Please prove by induction that for the nth
Chebyshev Polynomial of the First Kind T_{n}(x) in the variable x
with integer coefficients
[https://en.wikipedia.org/wiki/Chebyshev_polynomials],
defined by the linear recurrence T_{0}(x) = 1, T_{1}(x) = x,
T_{n+2}(x) = 2xT_{n+1}(x)  T_{n}(x) for all n ≥ 0 one has
T_{n}( (y + 1/y)/2 )
= (y^{n} + 1/y^{n})/2 for all n ≥ 0, real numbers y ≠ 0.

Due Thurs January 25, 11:59pm.
Consider the Catalan numbers as defined
C_{n} = binomial(2n,n) / (n+1)
in ENT, §1.2, Problem 10, page 12 for all n ≥ 0.
Prove that C_{n} = binomial(2n,n) – binomial(2n,n–1) for
all n ≥ 1, thus showing that the C_{n} are integers.

Due Tues January 30, 11:59pm.
ENT, §2.2, Problem 20(f), page 25:
If gcd(a, b) = 1, then gcd(a^{2}, b^{2}) = 1.
Hint: prove first that GCD(a,b^{2}) = 1,
and then GCD(A^2, B) = 1 for A=a and B = b^2;
or use the Fundamental Theorem of Arithmetic.

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.

Due Tues February 6, 11:59pm.
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–1.