Resources. Throughout the week, I've mentioned multiple resources that might interest you. I've attempted to group them by type/genre.

1. Light Reading. Novels and other sorts of things.
• Love and Math by Edward Frenkel tells about his journey with mathematics.
• The Man Who Loved Only Numbers by Paul Hoffman is a biography of the mathematician Paul Erdős.
• A Mathematician's Lament by Paul Lockhart is an essay about teaching mathematics as an art form.
• Arithmetic by Paul Lockhart
• You Look like a Thing and I Love You by Janelle Shane is a hilarious popular science book about artificial intelligence.
• The Music of the Primes by Marcus du Sautoy
• The City and the Stars by Arthur C. Clarke is a sci-fi novel from 1956. Clarke writes about the Ulam spiral (although he does not call it such). It's worth noting that Stanisław Ulam started studying his namesake spiral in 1963.
2. Biology. Interesting intersections of Biology & Number theory.
• A Biological Generator of Prime Numbers by Goles, Schulz, and Markus studies periodical insects. In particular, they study why certain types of cicadas appear every 7, 13, or 17 years.
• Are prime numbers special? Insights from the life sciences by Loconsole & Regolin is a survey on existing literature on prime numbers in the life sciences.
3. Music.
• Music by the Numbers by Eli Maor discusses how music has influenced mathematics.
• The Riemann Zeta Function and Tuning is an entire wiki about tuning, music theory, and math.
• The flaw in all western music is related to the Riemann Zeta function by Pieter Belmans is a short and interesting read about zeta functions and music.
4. Research Reading. Research papers, books, or other resources of this flavor.
• INTEGERS is an open access number theory journal.
• A Classical Introduction to Modern Number Theory by Ireland & Rosen is the sort of textbook you'd be likely to use in an upper division undergraduate course or early in grad school.

Project Ideas. Although I have a few precise topics listed below, I'm happy to explore projects related to links in the resources section as well!

1. In 1849, Polignac conjectured that every odd positive integer greater than 3 can be written in the form \(2^n+p\) for \(n\geq 1\) and \(p\) prime. This conjecture is false, (the smallest example is 127, found by Polignac himself) but paved the way for a lot of interesting mathematics. In 1950, Paul Erdős constructed infinitely many counterexamples to Polignac's conjecture.
   For this project, we'll be learning about covering systems of the integers, how Erdős used and constructed them, and how we can use them to prove the existence of numbers of a certain form. Questions of this flavor might look like:
• Are there infinitely many integers that cannot be written as the sum of a Fibonacci number and a prime? (The answer to this is yes, and was showing by Lenny Jones)
• Are there infinitely many composite sandwich numbers? Recall from our meeting(s) that a sandwich number is actually a sequence of numbers, like \(717, 7117, 71117, \dots\).
2. This week, we learned about integers modulo \(n\), but what about polynomials? Which values modulo \(n\) can be obtained by evaluating polynomials? For this project, we'll explore this!
3. Quadratic forms are polynomials where all terms are of degree 2, for example \(x^2+xy-y^2\). While seemingly simple, quadratic forms show up everywhere. (Even in very modern research-- like \(\mathbb{A}^1\)-homotopy theory!) There are a lot of things we could look at here:
• What values can quadratic forms take on modulo \(n\)?
• Fermat's theorem on sums of two squares determines when an integer can be written in the form \(x^2+y^2\), for \(x,y\in\mathbb{Z}\). We can also ask this question modulo \(n\)!
• Quadratic forms have a matrix associated to them. Does this work modulo \(n\)?