The TA for this course was Matthew Sunohara

What was the course about?

The course was about Diophantine Equations, that is, equations whose solutions must be integers. We covered, with varying degrees of depth, the following topics:

• Divisibility and the basics of congruences.
• Famous Diophantine Equations (Pell Equation, Markov Equations, Elliptic Curves, Fermat’s Last Theorem).
• Methods for producing solutions to Diophantine Equations.
• Methods for proving Diophantine Equations have no solutions.
• Combinatorial Methods (i.e. Ramsey Theory methods) to prove some equations have solutions.

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What motivated me?

What prompted me to pick this topic was to study the combinatorial aspect of solving linear equations when you paint the integers with colors. More concretely, assign a color to each integer and pick a linear equation, for example x + y = z, and ask how many integers you need to select to guarantee there is a monochromatic solutions. There are some ideas from Ramsey theory that guarantees that with enough numbers you will indeed produce these solutions.

Motivated by this, I decided to show many aspects of Diophantine Equations of which I had knowledge from other moments of life (olympiad and undergrad) but with a view toward how these are studied in more “serious” fashion.

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Exercises we worked on:

Note: The exercise section is not always complete. Many extra things were added on the spot as the course moved forward, depending on the questions of the public, and sometimes I do not have record of all of it.

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