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.

1. Using the algorithm discussed in class find a residue b modulo 53 such that b² ≡ 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).
2. 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.
3. ENT, §12.1, Problem 10, page 251.
4. ENT, §12.2, Problem 9, page 260. There are several solutions. One considers the Pythagorean triple (2y²,z,x²). 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.