MA-410 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 Feb 2, 11:55pm.
    Prove by induction that for the n-th Dickson Polynomial (for parameter –1) Dn(x) in the variable x with integer coefficients, defined by the linear recurrence
    D0(x) = 2, D1(x) = x, Dn+2(x) = xDn+1(x) + Dn(x) for all n ≥ 0
    one has
    Dn( y – 1/y  ) = yn + (–1)n/yn for all n ≥ 0, real numbers y ≠ 0.
  2. Due Tue Feb 2, 11:55pm.
    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 Thurs Feb 11, 11:55pm.
    ENT, §2.2, Problem 23, page 26: if a divides bc show that a divides gcd(a,b)gcd(a,c).
  4. Due Thurs Feb 11, 11:55pm.
    Please compute integers x,y,z such that 42x + 66y + 77z = 1. Please show your work, computing a solution to 42x + 66y = 6 and 6w + 77z = 1 first. Note: you only need to give one triple.
  5. Due Thurs Feb 18, 11:55pm.
    Prove for the extended Euclidean algorithm described in class: sk tk+1 – sk+1 tk = (–1)k+1 for all –1 ≤ k ≤ n–1.