MA-410 Homework 4

Due at 4:59pm in my mailbox in SAS 3151, Monday, April 30, 2012

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

  1. Using the algorithm discussed in class, find a residue b modulo 37 such that b2 ≡ 3 (mod 37). For the quadratic non-residue, use 2 modulo 37. Please show all your work (you may use Maple, but the required modular powers can be done by hand).
  2. ENT, §9.3, Problem 1(b), page 190. Please use Jacobi symbol reciprocity.
  3. ENT, §12.1, Problem 3, page 251.
  4. ENT, §12.2, Problem 9, page 260. Hint: a difficulty is that 2 may divide x (and therefore z) but not y, and then the Pythagorean triple (2y2, z, x2) is not easily made primitive in the same form. Handle that case separately by considering the equation modulo 16.