Math 58 - Number Theory

November 7, 2006

Homework 13 (due November 9):

About Diophantine equations

1. Find a few triples (x, y, z) of positive integers such that x² + y² = z⁴.

2. Prove or disprove: There is a triple of positive integers (x, y, z) such that x² + y² = z⁴⁰⁰.

3. Find all triples (x, y, z) of positive integers such that x² + 2y² = z².

(Possible approach: Imitate any of the methods we have used for x² + y² = z².)

About Gaussian integers

A Gaussian integer is a unit if it is 1, -1, i, or –i. These are exactly the Gaussian integers that have norm 1, and exactly the Gaussian integers that have multiplicative inverses.

A Gaussian integer x is a prime if it is not zero or a unit, and whenever it is written as a product x = y z, either y or z is a unit. For example, 7 is a prime. So are -7, 7i, and -7i. These are associated primes in the sense that any one of them can be gotten from any other by multiplying by a unit.

4. Is 1 + i prime? (Think of its norm. What would be the norms of its factors?)

5. Is 2 (that is, 2 + 0i) prime?

6. Is 3 prime?

7. Is 5 prime?

(end)