189-346/377B: Number Theory Assignment 6

Due: Monday, April 4

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. By elementary arguments (working in Z) show that the diophantine equation x^{2} + 1 = y^{n} has no solution when

1. x is odd and n > 1.

2. n is even.

Use this to show that if n > 1, then there exists a Gaussian integer a + bi for which x + i = (a + bi)^{n}. Conclude that b = ±1 and that the equation in question has no solution for n = 3, 5 and 7.

7. Solve the Pell equation x^{2} − 133y^{2} = 1 by using the continued fraction method (clearly indicate all the steps that you follow).

Math 377 students only:

8. Section 8.4, Problem 4 in Leveque.

9. Section 8.4, Problem 5 in Leveque.