This spring, I taught Math 42: An Introduction to Elementary Number Theory at Brown University. An important aspect of the course was the final project. In these projects, students either followed up on topics that interested them from the semester, or chose and investigated topics related to number theory. Projects could be done individual or in small groups.
I thought it would be nice to showcase some excellent student projects from my class. Most of the projects were quite good, and some showed extraordinary effort. Some students really dove in and used this as an opportunity to explore and digest a topic far more thoroughly than could possibly be expected from an introductory class such as this one. With the students' permission, I've chosen five student projects (in no particular order) for a blog showcase (impressed by similar sorts of showcases from Scott Aaronson).
Factorization Techniques, by Elvis Nunez and Chris Shaw. In this project, Elvis and Chris look at Fermat Factorization, which looks to factor $n$ by expressing $n = a^2 - b^2$. Further, they investigate improvements to Fermat's Algorithm by Dixon and Kraitchik. Following this line of investigation leads to the development of the modern quadratic sieve and factor base methods of factorization.
Pseudoprimes and Carmichael Numbers, by Emily Riemer. Fermat's Little Theorem is one of the first "big idea" theorems we encounter in the course, and we came back to it again and again throughout. Emily explored the Fermat's Little Theorem as a primality test, leading to pseudoprimes, strong pseudoprimes, and Carmichael numbers. [As an aside, one of her references concerning Carmichael numbers were notes from an algebraic number theory class taught by Matt Baker, who first got me interested in number theory].
Continued Fractions and Pell's Equation, by Max Lahn and Jonathan Spiegel. As it happened, I did not have time to teach continued fractions in the course. So Max and Jonathan decided to look at them on their own. They explore some ideas related to the convergence of continued fractions and see how one uses continued fractions to solve Pell's Equation.
Quantum Computing, by Edward Hu and Chris Long. Edward and Chris explore quantum computing with particular emphasis towards gaining some idea of how Shor's factorization algorithm works. For some of the more complicated ideas, like the quantum Fourier transform, they make use of heuristic and analogy to purvey the main ideas.
Fermat's Last Theorem, by Dylan Groos, Natalie Schudrowitz, and Kenneth Berglund. Dylan, Natalie, and Kenneth provide a historical look at attacks on Fermat's Last Theorem. They examine proofs for $n=4$ and Sophie Germaine's remarkable advances. They also touch on elliptic curves and modular forms, hinting at some of the deep ideas lying beneath the surface.
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