MA410, Spring 2024, Homework 4,
due as indicated for each problem.
All solutions must be submitted on the Moodle web site
for the class at
wolfware.ncsu.edu.
You may upload a photo of your handwritten solution or
a file of your typed solution.
Note my office hours on my
schedule.

Due Thursday, April 18, 11:59pm.
Using the
TonelliShanks Algorithm
discussed in class
find a residue b modulo 137
such that b^{2} ≡
2 (mod 137). For the quadratic nonresidue,
please use 3 ∈ ℤ_{137}.
Please use Maple /
wolframalpha.com
and show which modular powers
r &^ e mod 137 / PowerMod[r,e,137]
and modular products you have computed.
Note: 137 ≡ 9 (mod 16) so the formula from Homework 3, Problem 5
could be used for the squareroots of quadratic residues a modulo 137.
The problem here uses the algorithm that works for all primes.

Due Tuesday April 23, 11:59pm.
ENT, §12.2, Problem 9, page 260.
Prove that the Diophantine equation
x^{4} – 4 y^{4} = z^{2}
has no solution in positive integers x, y, z > 0.
[Hint in book:
Rewrite the given equation as (2y^{2})^{2}
+ z^{2} = (x^{2})^{2}
and appeal to Theorem 12.1 (every primitive Pythagorean
triple x, y, z with x even is of the form x = 2st,
y = s^{2} – t^{2},
z = s^{2} + t^{2}.)]
[Additional hint by me: Note that GCD(s,t) = 1.
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.
There is a second, very short solution on the Internet.]