## MA-410 Homework 3

Due at 4:59pm in my mailbox in SAS 3151,
Tuesday, April 11, 2017

Solutions may only be submitted in hard copy.
Note my office hours on my
schedule.

- ENT, §6.2, Problem 4(b), page 116.
- ENT, §8.4, Problem 2, page 167.
Please use the residue 2 for the primitive root.
- ENT, §9.1, Problem 8, page 174.
Please prove part (a) for r being a quadratic non-residue.
- ENT, §10.1, Problem 14, page 209.
[Hint: use Maple's “&^ mod” procedure.]
- Bonus problem:
Let p be a prime ≡ 5 (mod 8);
then p-1 ≡ 0 (mod 4) and p+3 ≡ 0 ≡ 3p+1 (mod 8).

Let a be a quadratic residue, and r a quadratic non-residue,
and let b = 2^{-1}
((1+r^{(p-1)/4}) a^{(3p+1)/8}
+ (1-r^{(p-1)/4}) a^{(p+3)/8}
) mod p. Please prove that b^{2} ≡ a (mod p).