#6: The Sum of Two Squares

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.

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#3: Rings

The point of this post is to provide examples and non-examples for the following ring classes:

  1. Ring
  2. Commutative Ring
  3. Integral Domain
  4. Integrally Closed Domain
  5. Unique Factorization Domain (UFD)
  6. Principal Ideal Domain (PID)
  7. Euclidean Domain
  8. Field

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