#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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Works

I have added a page on the menu bar called Works. It simply lists some of the mathematical works I have done, and I intend to keep it as an updated math portfolio. The primary addition is my paper on the Riemann zeta function that I wrote last Spring for a complex analysis course, and also its corresponding presentation.

Analytic Number Theory and Other Updates

Over the winter break I self-studied Tom Apostol’s Introduction to Analytic Number Theory book and completed the first three chapters, as well as the majority of the fourth. I posted the solutions, in pdf format this time, to the Solutions page. I chose this subject because Cornell currently does not have any course on analytic number theory, though it does have a graduate course in algebraic number theory, which I am taking this semester. More of that information is in a post on my main blog.

As stated in that post, I am now a double major in mathematics and computer science. This semester I am also planning to take all my notes in real-time LaTeX, aka LiveTeXing, as someone pointed out. The notes I have uploaded to Scribd, as it is more organized there and easier to view at a glance. They should be uploaded by the end of the same day of each lecture. I may upload them directly to this blog as well for centralization.

Courses that I am LiveTeXing: CS 4820, CS 4850, Math 4340, Math 7370.

I am also taking CS 3410, but the professor is posting all the powerpoints online, so it would be redundant to LiveTeX it.

#5: The Cantor Set and the Cantor Function

The Cantor set C is a fractal that is obtained by repeatedly removing the middle third of a segment. Start with the closed interval [0,1]. Remove the open interval (\frac{1}{3},\frac{2}{3}) to obtain [0,\frac{1}{3}] \cup [\frac{2}{3},1], i.e. two disjoint closed segments. Remove the middle thirds of those two segments, and you end up with four disjoint segments. After infinitely many steps, the result is called the Cantor set. The diagram below is only an approximation after a finite number of steps.

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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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