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.

1. ENT, §6.2, Problem 4(b), page 116.
2. ENT, §8.4, Problem 2, page 167. Please use the residue 2 for the primitive root.
3. ENT, §9.1, Problem 8, page 174. Please prove part (a) for r being a quadratic non-residue.
4. ENT, §10.1, Problem 14, page 209. [Hint: use Maple's “&^ mod” procedure.]
5. 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 b² ≡ a (mod p).