MA-410 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 Tuesday April 27, 11:55pm.
    Problem 2 on the Spring 2020 Exam.
  2. Due Tuesday April 27, 11:55pm.
    Using the Tonelli-Shanks Algorithm discussed in class find a residue b modulo 41 such that b2 ≡ 2 (mod 41). For the quadratic non-residue, use 15 ∈ ℤ41. Please show all your work (you may use Maple/wolframalpha.com, but the required modular powers could be done by hand).
  3. Due Thursday April 29, 11:55pm.
    ENT, §12.1, Problem 2, page 251: If x, y, z is a primitive Pythagorean triple, prove that x + y and x – y are congruent modulo 8 to either 1 or 7.
  4. Due Thursday April 29, 11:55pm.
    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.
    [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: 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.]