## MA-410, 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.

1. Due Thursday, April 18, 11:59pm.
Using the Tonelli-Shanks Algorithm discussed in class find a residue b modulo 137 such that b2 ≡ 2 (mod 137). For the quadratic non-residue, please use 3 ∈ ℤ137. 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.
2. Due Tuesday April 23, 11:59pm.
ENT, §12.2, Problem 9, page 260. Prove that the Diophantine equation x4 – 4 y4 = z2 has no solution in positive integers x, y, z > 0.
[Hint in book: Rewrite the given equation as (2y2)2 + z2 = (x2)2 and appeal to Theorem 12.1 (every primitive Pythagorean triple x, y, z with x even is of the form x = 2st, y = s2 – t2, z = s2 + t2.)]
[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.]