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Number Theory and Combinatorics

This page contains a list of ideas for DRP projects, but is by no means exhaustive.

Topic: Generating Functions
Suggested Text: generatingfunctionology, Herbert S. Wilf
Suggested Background: MATH 1301 (Accelerated Single-Variable Calculus II)
Description: Using the idea of Taylor series but only requiring basic algebra, generating functions prove to be one of the most effective methods of understanding sequences of numbers in combinatorics and number theory. For example, there is a simple (non-recursive) formula for the Fibonacci numbers that one can very quickly prove! Generating functions give wide-ranging methods that allow one to prove surprising binomial and combinatorial identities that would be otherwise difficult to prove or even discover in the first place.

Topic: Irrationality and Transcendence
Suggested Text: An Introduction to Number Theory, Ivan Niven, Herbert Zuckerman, & Hugh Montgomery, and Transcendental Numbers, M. Ram Murty & Purusottam Rath
Suggested Background: MATH 3300 (Abstract Algebra) is a prerequisite for the more advanced transcendence theory material
Description: Irrational numbers are real numbers which cannot be expressed as the quotient \(n/m\) of two integers (where \(m\) is non-zero). Transcendental numbers are real numbers which are not the roots of any polynomial of a single-variable whose coefficients are rational. Famously, both \(e\) and \(\pi\) are transcendental, but proving this is extremely non-trivial. In general, transcendentality is difficult to prove (it is known that at least one of \(e+\pi\) and \(e\cdot \pi\) are transcendental, but neither has been proven so). However, it ends up that almost all real numbers are transcendental!

Topic: Basic Analytic Number Theory
Book: Introduction to Analytic Number Theory, Tom Apostol
Suggest Background: MATH 1301 (Accelerated Single-Variable Calculus II)
Description: One of the fundamentally fascinating aspects of number theory is the interplay between the discrete and the continuous. While it is difficult to determine when exactly a prime number occurs, the prime number theorem tells us that the number of primes less than or equal to \(x\) is approximately \(x/\log x\) for \(x\) sufficiently large. This tells us that although many of the quantities we may wish to know about prime numbers (such as how many there are up to a given point) may be better studied by understanding continuous approximations to them. As a result, we can use basic calculus techniques to understand prime numbers much better.

Topic: Elliptic Curves
Book: Rational Points on Elliptic Curves, Joseph H. Silverman & John Tate
Suggested Background: MATH 3300 (Abstract Algebra), MATH 3800 (Theory of Numbers)
Description: Elliptic curves have become one of the most exciting fields of study in recent years. Fundamentally, elliptic curves are simply the solution set of \(y^2 = x^3 + ax + b\), which would appear to not be much more difficult to understand than conic sections. However, it turns out that they contain a breadth of number theoretic information, being fundamental to Andrew Wiles’s proof of Fermat’s Last Theorem. Additionally, they have proved useful in a variety of other areas, such as cryptography. A reading program in this area would entail learning the basics, giving the student an understanding of both the difficulty and depth of this area, and allowing them to see why so many mathematicians have become fascinated with these objects.

Topic: \(p\)-adic Analysis
Suggested Text: \(p\)-adic Numbers – An Introduction, Fernando Q. Gouvea
Suggested Background: MATH 3100 (Introduction to Analysis) or MATH 3300 (Abstract Algebra)
Description: \(p\)-adic analysis represents a different approach to correcting the failings of the field of rational numbers, with a resulting theory that looks wildly different from the classical analysis of the real line. The \(p\)-adic reals (for each prime \(p\)) are nevertheless rich objects of study, both from an analytic point of view, as well as an algebraic one.

Topic: Probabilistic Methods in Combinatorics and Number Theory
Book: The Probabilistic Method, Noga Alon & Joel H. Spencer
Suggested Background: Familiarity with graph theory
Description: Using basic probability, it is possible to prove a number of results in graph theory and number theory that would be extremely difficult to prove otherwise. In order to show that certain types of graphs exist for example, one could directly construct such graphs; however, by making a collection of graphs into a probability space and proving that such a graph exists with non-zero probability, this existence proof may only take a few lines. One of the reasons for Erdős’s impressive mathematical abilities is the mastery of this method. In a directed reading program, a student would be able to come familiar with the basic ideas of this technique and gain some understanding as to when certain problems may be particularly susceptible to it.