MA-410 Spring 2023 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 Tue January 24, 11:55pm.
    Please prove by induction that for the n-th Dickson Polynomial (for parameter 2) Dn(x) in the variable x with integer coefficients, defined by the linear recurrence
    D0(x) = 2, D1(x) = x, D2(x) = x2 – 4,     Dn+2(x) = x Dn+1(x) – 2 Dn(x) for all n ≥ 0
    one has
    Dn( y + 2/y  ) = yn + (2/y)n for all n ≥ 0, real numbers y ≠ 0.
  2. Due Tue January 24, 11:55pm.
    Consider the Catalan numbers which can be defined inductively as C0 = 1, Cn = (4n–2)/(n+1) Cn–1 for all n ≥ 1 (cf. ENT, §1.2, Problem 10, page 12). Please prove that Cn = binomial(2n,n) – binomial(2n,n–1) for all n ≥ 1, thus showing that the Cn are integers.
  3. Due Tue Jan 31, 11:59pm.
    ENT, §2.2, Problem 23, page 26: if a divides bc show that a divides gcd(a,b)gcd(a,c).
  4. Due Tue Jan 31, 11:59pm.
    Please compute integers x,y,z such that 72x + 88y + 99z = 1. Please show your work, computing a solution to 72x + 88y = 8 and 8w + 99z = 1 first. Note: you only need to give one triple.
  5. Due Tuesday Feb 7, 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.