Stanford University EPGY
Number Theory
Number Theory
Stanford University EPGY Summer Institutes 2010
Suggested Final Presentation Topics
This list contains possible final presentation topics for your group presentations. However, you are
not limited to the topics in this list. Feel free to discuss other possible topics (e.g. extensions of
topics and/or problems that we have studied in class so far, number theory topics that you have
read about in other courses, etc.) with me and/or with the teaching assistants! When choosing
a topic, keep in mind that your presentation should last approximately 20-30 minutes, and you
should design a presentation that is clear, focused, and interactive. Please consult with me and/or
the teaching assistants to design, print, and make copies of any worksheets or other materials that
you’ll need for your presentation. Some useful websites for mathematical research in number theory
are listed on the back of this page.
• The Chinese Remainder Theorem and its applications; linear congruence equations;
1,2
• Primitive roots1,2
• Squares modulo p, quadratic residues, quadratic reciprocity, and the Legendre symbol1,2
• Mersenne primes, Fermat primes, and perfect numbers1,2
• RSA public key cryptography1,2
• Classification (and properties) of Pythagorean triples1,2
• Which primes/numbers are sums of two squares? Sums of three squares?1,2
• Lagrange’s Theorem and sums of four squares1,2
• Magic squares1,4 , De la Loubere method1 , Siamese method4 , Conway’s LUX method4 , magic
cubes4
• Transcendental numbers3,4 and Liouville’s theorem; see also http://www.math.sc.edu/ filaseta/gradcourses/Math785/notes785.html for lecture notes from the University of South
Carolina on this topic. In particular, the first 6 files (through “The Transcendence of e and
π”) will be particularly useful. Please let me know if you need me to print copies of these
notes for you.
• Gaussian integers2,3,4
• Distribution of primes2
• Pseudoprimes and Carmichael numbers2,4
• Arithmetic functions, the Mobius Inversion Formula, and applications2
1 Use
the course reader (and other sources) as a reference.
the the course textbook, Elementary Number Theory, Gareth A. Jones and Josephine M. Jones (and other
sources) as a reference.
3 Reference information in An Introduction to the Theory of Numbers, G.H. Hardy and E.M. Wright (see the
teaching assistants to look at topics in this textbook; we can photocopy portions of this book for you as necessary).
4 Use internet references and/or the mathematics library to obtain reference information. We can check out books
from the Stanford mathematics library for you to use as references. See http://socrates.stanford.edu.
2 Use
paquin@math.stanford.edu
1
Number Theory
Stanford University EPGY
Number Theory
• The Riemann Zeta function2
• Minkowski’s Theorem and its applications (e.g. a proof of the Two Squares Theorem using
Minkowski’s Theorem)2,3
• Fermat’s Last Theorem2
• Ducci sequences and the length of the Four Numbers Game4
• Isosceles right triangles and irrationality; Fermat’s method of descent1,2
• Additional proofs that there are infinitely many primes; Proofs that there are infinitely many
primes of the form 4n + 3, 6n + 5, 8n + 5, etc.3,4
• Unsolved problems concerning primes (e.g. the twin prime conjecture, Goldbach’s conjecture,
etc.)3,4
• Primality testing2,3,4
• Bachet’s problem of the weights3,4
• Approximation of irrationals by rationals3,4
• Detailed discussion of an unsolved problem in number theory (see me if you would like to
borrow a copy of the book Unsolved Problems in Number Theory)
Some useful websites for internet research (note that this is only a partial list; there
are numerous other sites not on this list that you can use as a reference for your final
presentation).
1. Wikipedia (note–make sure that you completely verify any information that you obtain from
Wikipedia!): http://www.wikipedia.org
2. Wolfram MathWorld: http://mathworld.wolfram.com
3. Number Theory Web: http://www.numbertheory.org
4. Online lecture notes on the Number Theory Web:
http://www.numbertheory.org/ntw/lecturenotes.html
5. The Prime Pages: http://primes.utm.edu
6. Great Internet Mersenne Prime Search (GIMPS): http://www.mersenne.org
7. Elementary Number Theory, by W. Edwin Clark (an online textbook):
http://shell.cas.usf.edu/∼wclark/elemnumthbook.pdf
8. Definitions, Solved and Unsolved Problems, Conjectures, and Theorems in Number Theory
and Geometry, edited by M.L. Perez (an online textbook):
http://fs.gallup.unm.edu//Definitions-book.pdf
1 Use
the course reader (and other sources) as a reference.
the the course textbook, Elementary Number Theory, Gareth A. Jones and Josephine M. Jones (and other
sources) as a reference.
3 Reference information in An Introduction to the Theory of Numbers, G.H. Hardy and E.M. Wright (see the
teaching assistants to look at topics in this textbook; we can photocopy portions of this book for you as necessary).
4 Use internet references and/or the mathematics library to obtain reference information.
2 Use
paquin@math.stanford.edu
2
Number Theory