Math 168: Homework Assignment 1

William Stein

Due: Wednesday, Oct 5, 2005

The problems have equal point value, and multi-part problems are of the same value. You are allowed to use a computer on any problem, as long as you include the exact code used to solve the problem with your solution. Any software systems (e.g., Magma, SAGE, Mathematica, C) are allowed.

1 Announcements

1. Homework will typically be assigned on Mondays and due on the following Wednesday. Then you have plenty of time to ask me questions about it.

2. Grading: For undergrads it is exactly as on the syllabus. For graduate students, make some effort on the homework and do a final project and you will get an A. I will greatly appreciate if grad students help the undergraduates in the course.

2 Problems

1. Prove that there are infinitely pairs (x, y) of rational numbers such that 3x^{2} + 4y^{2} = 7.

2. Find (by brute force) all pairs (x, y) of integers with 0 ≤ x < 5 and

0 ≤ y < 5 and

y^{2} ≡ x^{3} − x (mod 5).

3. Find an elliptic curve E over a finite field 𝔽_{p} such that the group E(𝔽_{p}) is not cyclic.

4. I’m giving you an account on my “super-fast” dual-opteron server modular.ucsd.edu, where you’ll be able to run SAGE and Magma. Please select a login name. See also http://modular.ucsd.edu/calc/.

5. Find 10 distinct solutions (x, y), with x, y ∈ ℚ to the equation

y^{2} + y = x^{3} − 7x + 6.