MA410 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.
 Using the TonelliShanks Algorithm discussed in class
find a residue b modulo 41
such that b^{2} ≡
21
(mod 41). For the quadratic nonresidue,
use 3 ∈ ℤ_{41}. Please show all your work (you may use Maple, but the required
modular powers could be done by hand).

Problem 2 on the
Spring 2016 Exam.
 ENT, §12.1, Problem 4, page 251.
 ENT, §12.2, Problem 9, page 260.
There are several solutions. One considers the Pythagorean triple
(2y^{2},z,x^{2}). 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.