Randomness in Number Theory

Faculty Member

A J Hildebrand

Project Image

Random-like behavior is ubiquitous in number theory. For example, the prime numbers, the digits of pi and other famous constants, and the values of the Moebius function and similar number-theoretic functions, all appear to behave much like appropriately defined "true" random sequences. In this project we seek to explore such random features experimentally - via large scale computations and geometric visualizations as random walks - and if possible also theoretically. The specific topic(s) to be explored this semester will depend on the background and interests of the participants and will be finalized at the beginning of the semester. Past editions of this project have focused on random walks associated with the Moebius function, quadratic residues, and the digits of Pi and other irrational numbers. See http://www.math.illinois.edu/~ajh/ugresearch/wolfram-demonstrations.html for examples of visualizations resulting from past projects that have been published as "Wolfram Demonstrations" at http://www.demonstrations.wolfram.com. Mathematica will be the primary tool for the experimental component this project. Prior experience with Mathematica is desirable, though not necessary.

Team Meetings

Weekly

Project Difficulty

Intermediate

Undergrad Prerequisites

There are no hard prerequisites, though completion of Math 453 (or an equivalent number theory course) with an A grade, or other evidence of a strong mathematical background such as an A grade in an honors-level or proof-based math course would be desirable. Prior experience with Mathematica or evidence of very strong general coding skills would be desirable, but is not necessary.