MA-410 Homework 4

Due at 4:59pm in my mailbox in SAS 3151, Thursday, April 25, 2019


Solutions may only be submitted as paper documents. Note my office hours on my schedule.

  1. Using the Tonelli-Shanks Algorithm discussed in class find a residue b modulo 41 such that b2 ≡ 21 (mod 41). For the quadratic non-residue, use 3 ∈ ℤ41. Please show all your work (you may use Maple, but the required modular powers could be done by hand).
  2. Problem 2 on the Spring 2016 Exam.
  3. ENT, §12.1, Problem 4, page 251.
  4. ENT, §12.2, Problem 9, page 260. There are several solutions. One considers the Pythagorean triple (2y2,z,x2). In order for Theorem 12.1 to apply, the triple has to be primitive. A difficulty is that if 2 divides x (and therefore z) but not y, one cannot shrink the size of the triple keeping its form. One can handle that case by considering the equation modulo 16.