## MA-410 Homework 4

Due at 4:59pm in my mailbox in SAS 3151,
Thursday, April 27, 2017

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 53
such that b
^{2} ≡ 7 (mod 53). For the quadratic non-residue,
use 5 modulo 53. Please show all your work (you may use Maple, but the required
modular powers could be done by hand).
- ENT, §9.3, Problem 1(c), page 190.
Please use Jacobi symbol reciprocity without factoring numerator or denominator
except removing powers of 2 from the numerator.
- ENT, §12.1, Problem 10, 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.