Math 168: Homework Assignment 4

William Stein

Due: Wednesday, Oct 26, 2005

The problems have equal point value, and multi-part problems are of the same value.

1 Problems

1. Use the enumeration of Pythagorean triples (a, b, c) with a^{2}+b^{2} = c^{2} to create a program that lists all square-free congruent numbers. These will, of course, not be in increasing order! You do not know when a given congruent number will appear on your list.

2. In this problem you will create a procedure to output all triangles with rational side lengths and area 219.

(a) Let E be the elliptic curve y^{2} = x^{3} − 219^{2}x. Compute generators for the Mordell-Weil group E(ℚ) using a computer program (e.g., SAGE, Magma, mwrank, etc.) [Hint: E has rank 2.]

(b) Describe how to systematically list all elements of E(ℚ) with y ≠ 0.

(c) Use Prop 6.5.7 from the class notes and your answer to the above question to systematically list all triangles with rational side lengths and area 219.