The following gives a list of possible topics for your number theory project. You are also free to come up with your own choice of topic related to the course.

Your write-up should be about 10-15 pages in length. It is OK, albeit not required, to include computer programs and the outcomes of numerical experiments in your project. The projects are meant to be carried out individually, not in teams, although you are free (in fact, encouraged) to discuss what you are working on with your classmates.

You should choose a project topic, and let me know what it is, before the week of the Spring break at the end of February. To avoid duplications, I will be posting your choices on the web, as they are made. Project topics will be assigned on a first-come, first serve basis: I expect that some of the more popular topics will be chosen first, so it is in your interest to make a fast decision!

You may pick one of the chapters (between ch. 2 and ch. 8) in the book by Scharlau and Opolka and write a project on the mathematician that is featured in that chapter, giving more details to flesh out the mathematics alluded to in the text.

Explain the Miller-Rabin primality test and Miller's analysis of its complexity, based on the (generalized) Riemann hypothesis.

Miller, Gary L. Riemann's hypothesis and tests for primality. Working papers presented at the ACM-SIGACT Symposium on the Theory of Computing (Albuquerque, N.M., 1975). J. Comput. System Sci. 13 (1976), no. 3, 300--317.

Carl Pomerance, Recent developments in primality testing, Mathematical Intelligencer 3, no. 3 (1981) 97-105.

Manindra Agrawal, Neeraj Kayal, Nitin Saxena, PRIMES is in P. Annals of Mathematics 160(2): 781-793, 2004. (Or, you can dowload the original paper directly.)

Lenstra, H. W., Jr. Factoring integers with elliptic curves. Ann. of Math. (2) 126 (1987), no. 3, 649--673.

Koblitz, Neal. A course in number theory and cryptography. Second edition. Graduate Texts in Mathematics, 114. Springer-Verlag, New York, 1994.

Recently a probabilistic algorithm was discovered for factoring integers in polynomial time on a

Peter Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, SIAM Journal on Computing Volume 26, Number 5 pp. 1484-1509.

Serre, J.-P. A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.

Davenport, Harold. Multiplicative number theory. Second edition. Revised by Hugh L. Montgomery. Graduate Texts in Mathematics, 74. Springer-Verlag, New York-Berlin, 1980.

Khintchine, A. Ya. Continued fractions. Translated by Peter Wynn. P. Noordhoff, Ltd.,

Dunham, William. Journey through genius. The great theorems of mathematics. Penguin Books, New York, 1991.

Koblitz, Neal. p-adic numbers, p-adic analysis, and zeta-functions. Second edition. Graduate Texts in Mathematics, 58. Springer-Verlag, New York-Berlin, 1984.

Iwasawa, Kenkichi. Lectures on p-adic L-functions. Annals of Mathematics Studies, No. 74. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972.

van der Poorten, Alfred. A proof that Euler missed... Apery's proof of the irrationality of zeta(3). An informal report. Math. Intelligencer 1 (1978/79), no. 4, 195--203. Groningen 1963.

Andrews, George E. The theory of partitions. Reprint of the 1976 original. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998.

Schoof, Rene. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp. 44 (1985), no. 170, 483--494.

Turing, Alan. M. Some calculations of the Riemann zeta-function. Proc. London Math. Soc. (3) 3, (1953). 99--117.

Odlyzko, Andrew M. Analytic computations in number theory. Mathematics of Computation 1943--1993: a half-century of computational mathematics (Vancouver, BC, 1993), 451--463, Proc. Sympos. Appl. Math., 48, Amer. Math. Soc., Providence, RI, 1994.

Koblitz, Neal p-adic numbers, p-adic analysis, and zeta-functions. Second edition. Graduate Texts in Mathematics, 58. Springer-Verlag, New York-Berlin, 1984.

Gouvea, Fernando Q. p-adic numbers. An introduction. Second edition. Universitext. Springer-Verlag, Berlin, 1997.

Ireland, Kenneth; Rosen, Michael. A classical introduction to modern number theory. Second edition. Graduate Texts in Mathematics, 84. Springer-Verlag, New York, 1990.