## Course Summary

Number theory has evolved through several stages in the past two millennia. Notions of primality and divisibility are indeed quite classical, and the ancients even knew a great deal about some relatively sophisticated ideas: the infinitude of primes, perfect numbers, etc. Innovation sat stagnant for about 15 centuries until, around 1625, Fermat acquired a copy of *Diophantus* and started pursuing the subject in earnest. Even this work might have fallen to the wayside had Euler not continued Fermat's work about 100 years later. Since that time, many mathematical greats have made significant contributions to the subject and discovered surprising connections between number theory and almost all other branches of mathematics. Perhaps the most striking feature of modern (i.e., post-Fermat) number theory is the divide between the simplicity of theorem statements and the complexity of techniques used to prove them.

In this class, we'll focus on answering the following question: for an integer $n$, how can we characterize those primes $p$ for which there exist integers $x$ and $y$ with $p=x^2+ny^2$. As a follow-up to 305, we'll be able to use the power of abstract algebra to answer this question, and we'll see that this seemingly innocuous problem requires some sophisticated machinery to unravel. One of interesting trends will uncover is the increased complexity of machinery necessary to resolve this problem as the value of n increases.

## Course Instructor

The professor for this class is Andy Schultz. His office is on the third floor of the Science Center, room S352. His office hours are Mondays from 1-3, Wednesdays from 10-11, and Thursdays from 3-4. You are highly encouraged to attend office hours, and you never need an appointment to do so. If these office hours don't fit with your schedule, contact the instructor so that he can either adjust when ``official" office hours are held or set up an appointment to help you outside of office hours.

You can contact the instructor at . Though he is always happy to receive emails from you with questions or concerns about the course, he can't guarantee that he'll be able to promptly reply to emails late at night or over the weekend. If you do contact the professor by email, please be sure to follow standard email etiquette. In particular, please make sure you include a greeting and signature and avoid abbreviations. If you're contacting him to ask about a problem, please be sure to specify what the problem asks (as opposed to asking something like ``I can't get problem 2 and need your help").