189-346/377B: Number Theory Assignment 6

Due: Monday, March 26

1. Let p be an odd prime. Show that −2 is a quadratic residue modulo a prime p if and only if p is a prime of the form m^{2} + 2n^{2}.

2. Use question 1 and quadratic reciprocity to get a complete characterisation of all the integers that are of the form m^{2} + 2n^{2}.

3. Repeat questions 1 and 2 with m^{2} + 2n^{2} replaced by m^{2} + 3n^{2}.

4. Show that there are primes p for which −5 is a quadratic residue modulo p, yet which are not of the form m^{2} + 5n^{2}.

5. Make a list of the integers ≤ 100 that can be written in the form m^{2} +5n^{2}, and 2m^{2} + 2mn + 3n^{2}. Can you formulate some conjectures about how these sets of integers behave? (You may find it useful to write each integer in factored form.)

6. Section 7.3, Problem 7 in Leveque.

7. Section 8.4, Problem 4 in Leveque.

377:

8. Section 8.4, Problem 5 in Leveque.