MAT 465 -- Theory of Numbers
Course Information
Professor: D. Bradley
Website: http://www.umemat.maine.edu/faculty/bradley/index.html
Although this is a senior-level course, the nature of the subject is such that prerequisites are minimal. In contrast with many other branches
of mathematics, students of number theory find that they can readily
understand and appreciate the questions lying at the frontiers of current
knowledge with very little background. We'll use Hardy and Wright's
classic text as a springboard for discussing topics ranging from as yet unanswered questions raised by the ancient Greeks, to Fermat's Last Theorem and modern applications to cryptography, digital signatures, watermarking,
and coding theory.
Text:
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, (5th ed.) Clarendon Press, Oxford, 1990.
This extremely popular and otherwise excellent book lacks a subject index. But you can get one by clicking here: Subject Index
for Hardy & Wright
References:
- Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery, An Introduction to the Theory of Numbers, John Wiley, New York, 1991.
- Godfrey Harold Hardy, A Mathematician's Apology, Cambridge University Press, 1992.
- Paulo Ribenboim, The Book of Prime Number Records, (2nd ed.)
Springer-Verlag, New York, 1989.
Niven, Zuckerman and Montgomery would make a reasonable alternative text. It has a good selection of exercises.
We'll read and discuss portions of Hardy's famous Apology in class
for an occasional change of pace.
In reference to Ribenboim's record book above, see the poster on my office door, courtesy of coauthor and reknowned computational number theorist Richard Crandall. The poster displays in micro-print the digits of the largest (as of Jan. 1998)** explictly known prime number, 23021377-1.
Syllabus: I plan to cover at least some of the following topics:
- Prime Number Theory
- fundamental theorem of arithmetic
- Euclidean algorithm
- proofs there are infinitely many primes
- Bertrand's postulate, Mertens theorem
- primes in arithmetic progressions
- prime number theorem, sieves
- perfect numbers, Mersenne primes
- Goldbach's conjecture, twin prime conjecture
- Arithmetic Functions
- Euler's totient function
- Möbius function, Möbius inversion
- Dirichlet's divisor problem
- generating functions and Dirichlet series, Riemann's zeta function
- Ramanujan's sum
- Congruences and Residues
- Fermat's little theorem, Wilson's theorem
- applications to cryptography
- primality testing, Carmichael numbers
- quadratic reciprocity
- Farey fractions
- Bernoulli numbers, von Staudt's theorem
- Irrational Numbers
- irrationality proofs
- algebraic and transcendental numbers, normal numbers
- e and p
- Liouville's theorem, Roth's theorem
- continued fractions
- Diophantine Equations
- Pythagorean triples
- Pell's Equation, Archimedes' Cattle Problem
- Fermat's Last Theorem
- Partitions
- Jacobi's Triple Product
- Rogers-Ramanujan identities
- Ramanujan's continued fraction
- Waring's Problem
- sums of squares
- theta functions
Number theory is a huge subject with many branches and a vast literature. Click here for a separate page with standard references in each of the major areas. I've also listed a few of the hundreds of number theory-related internet sites below.
Internet Resources:
Help me add to this list by suggesting additional number theory related links!
*Of course, you are welcome to drop
by the office any time, or make an appointment.
**This record has since been superceded.
Check out the Great Internet Mersenne Prime Search at http://www.mersenne.org/ or
http://www.utm.edu/research/primes/notes/13466917/. Richard Crandall has once again made a poster containing the complete printout of the over four million digits in the new record-breaking prime.