MATH 319 Exploratory Assignment 3 – Primes of the form x² + ny²

(Due Wednesday, February 26)

We will prove in class that a prime p can be written as x² + y² for integers x and y if and only if p = 2 or . Using the Maple program Primes of the form x²+ny² try to discover a congruence relation like the above for the case n = 2. If you find one, you might try to move to the case n = 3.

1. Find all the primes under 100 that are of the form x² + 2y².

2. Test them for congruence mod 4,5,6,7, and 8.

3. Look for a pattern.