#7: The Visible Grid Point Problem

A link to this article in pdf format: [pdf]

Here is a difficult probability question:

Suppose you are standing on an infinitely large square grid at the point (0,0), and suppose that you can see infinitely far but cannot see through grid points. Given a random grid point z = (x,y), where x and y are integers, what is the chance you can see z?

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#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.

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.