Pries: 405 Number Theory, Spring 2012. Homework 5.
Due: Friday 2/24.
Quadratic Reciprocity
Project: Pick a group (2-3 people) and your top two choices for a project topic. Hand
this in separately.
Read: Stein sections 4.1, 4.2, 4.3 (we only did case a = 2)
Do 6 of the following problems.
1. Stein 4.1, 4.3, 4.4.
2. Is there a solution to x2 + 14x − 35 ≡ 0 mod 337?
3. Suppose that a is a square modulo p and that p ≡ 1 mod 4. Show that x = a(p+1)/4 is
a solution to the equation x2 ≡ a mod p.
4. Prove there are infinitely many primes p congruent to 1 modulo 3. Hint: let N =
(2p1 · · · pr )2 + 3, show that there is at least one prime q dividing N such that q ≡
3 mod 4, then show q ≡ 7 mod 12.
5. Let q be a prime number such that p = 2q + 1 is also a prime number. Suppose
q ≡ 1 mod 4. Show that 2 is a primitive root modulo p.
A mathematician is a device for turning coffee into theorems. Paul Erdos
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