In the Algebraic Number Theory course that I am taking, we covered the problem of when a number is the sum of two squares. I wrote a short expository paper [pdf] to go with it. Here are the positive integers up to 15 which can and cannot be written as the sum of two squares:

Examples:

• 1 = 1² + 0²
• 2 = 1² + 1²
• 4 = 2² + 0²
• 5 = 2² + 1²
• 8 = 2² + 2²
• 9 = 3² + 0²
• 10 = 3² + 1²
• 13 = 3² + 2²

Non-examples:

• 3
• 6
• 7
• 11
• 12
• 14
• 15

Can you find the pattern? (Note: This is a very hard question. You may need to write down more examples to start noticing it.)

I have also posted some hints.