#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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#2: Epic Morphisms

Given that the name of this site is Epic Math, I felt I should write about a topic where the word epic is a technical term! It is hardly surprising that such a term exists, as many otherwise non-mathematical words such as almost, simple, open, connected, regular, normal, field, ringonto, map, twin, lucky, and even sexy have technical definitions.

Category Theory

The term epic is found in category theory, which is an extremely abstract branch of mathematics that formally deals with many other fields of math. It is so strange that some mathematicians have labeled it “abstract nonsense.”

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