MA410 Homework 3
Due at 4:59pm in my mailbox in SAS 3151,
Thursday, April 12, 2018
Solutions may only be submitted in hard copy.
Note my office hours on my
schedule.
 ENT, §6.2, Problem 7, page 116.
 ENT, §8.4, Problem 3, page 167.
 ENT, §9.1, Problem 8, page 174.
Please prove part (a) for r being a quadratic nonresidue.
 ENT, §10.1, Problem 15, page 209.
[Hint: use Maple's “&^ mod” procedure.]
 Bonus problem:
Let p be a prime ≡ 9 (mod 16);
then p1 ≡ 0 (mod 4) and p+7 ≡ 0 ≡ 7p+1 (mod 16).
Let r be a quadratic nonresidue, and let a be a quadratic residue.

Case a^{(p1)/4} ≡ 1 (mod p):
let b = x^{1}
(a^{(7p+1)/16} + a^{(p+7)/16}) mod p,
where x is the squareroot of 2 modulo p computed in
Problem 3 (Problem 8 on page 174 of the textbook).
Please prove that b^{2} ≡ a (mod p).

Case a^{(p1)/4} ≡ 1 (mod p):
let b = 2^{1}
((1+r^{(p1)/4})a^{(7p+1)/16}
+ (1r^{(p1)/4})a^{(p+7)/16}) mod p.
Please prove that b^{2} ≡ a (mod p).