Here's some more detail on some possible project ideas.

• Gauss sums A (quadratic) Gauss sum is a number like &Sigma_t=0..p-1 (t/p) &zeta^at where &zeta = exp(2&pi i/p) is a p^th root of unity and (t/p) is the Legendre symbol. Gauss sums have beautiful properties, and can be used to count solutions to Diophantine equations and to provide a proof of quadratic reciprocity.
  Possible references: [Ireland and Rosen, 6, 8] [Miller and Takloo-Bighash, 4.4] [Stein].
• Distribution of primes in congruence classes Towards the end of the course, we'll see that primes are essentially evenly distributed among those which are 1 mod 4 and those which are 3 mod 4. Here are two possible projects extending this idea:
  • Finer resolution Use a computer to gather numerical evidence about the distribution of primes modulo 4. Investigate what is known about these "higher-order" effects.
    Possible references: [Miller and Takloo-Bighash, 3.3] [Granville and Martin]
  • Other moduli One can also look at the distributions of primes modulo other numbers. In addition to the references cited above, see also [Pollack 2.2-2.6].
• Artin's Conjecture We'll see that for each prime p there is some number a whose order mod p is p-1, which is as large as possible. In the other direction, Artin conjectured that for any a there are infinitely many p such that the order of a is as large as possible (as long as a is neither -1 nor a perfect square). Do some computer experiments around this conjecture, and see what's known about the conjecture.
  Possible references: [Goldstein], [Silverman 20].
• Lagrange's four-square theorem While only some numbers can be written as a sum of two squares, every number can be written as a sum of four squares.
  Possible references: [Stillwell 8] [Farkas and Kra 7.3]

References

• L. Goldstein, Density questions in algebraic number theory, American Mathematical Monthly 78 (1971), 342-351.
• A. Granville and G. Martin, Prime number races, American Mathematical Monthly 113 (2006), no. 1, 1-33.
• K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag 1990.
• S.J. Miller and R. Talkoo-Bighash, An Invitation to Modern Number Theory, Princeton University Press 2006.
• P. Pollack, Not always buried deep.
• W. Stein, Elementary Number Theory.
• J. Stillwell, Elements of Number Theory, Springer-Verlag 2003.