## 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.

- Using the algorithm discussed in class, find a residue b modulo 37
such that b
^{2} ≡ 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).
- ENT, §9.3, Problem 1(b), page 190. Please use Jacobi symbol reciprocity.
- ENT, §12.1, Problem 3, page 251.
- 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
(2y
^{2}, z, x^{2}) is not easily
made primitive in the same form.
Handle that case separately by considering the equation modulo 16.