Math 480 (Spring 2007): Homework 8

Due: Monday, May 21

There are 6 problems. Each problem is worth 6 points and parts of multipart problems are worth equal amounts. You may work with other people and use a computer, unless otherwise stated. Acknowledge those who help you.

1. Write the integer $9000000000000000001053$ as a sum of two squares.

2. Evaluate the infinite continued fraction $[2,\overline{1,3}]$ . Your answer should be an explicit quadratic irrational number.

3. 1. Write down in any way (no proof required) the infinite continued fraction of the quadratic irrational number $\frac{1+\sqrt{7}}{2}$ . (Your answer should look like a finite continued fraction followed by a repeating part with a bar over it.)
   2. Prove that your answer to (a) is correct by doing algebra as in problem 2 to show that the value of the continued fraction you give is really $\frac{1+\sqrt{7}}{2}$ .)

4. Find a positive integer that has at least three different representations as the sum of two squares, disregarding signs and the order of the summands.

5. Let $E$ be the elliptic curve $y^2 = x^3 - 7x$ over the rational numbers.
   1. There is a point $P=(a,b)$ on $E$ with $a,b\in\mathbb{Z}$ and $\vert a\vert < 10$ . Find it.
   2. Compute $Q = P+P$ by any method.

6. Let $E$ be the elliptic curve $y^2 = x^3 + 2x$ over the finite field $\mathbb{Z}/3\mathbb{Z}$ .
   1. Show that $\char93 E(\mathbb{Z}/3\mathbb{Z}) = 4$ , i.e., that there are $3$ solutions to $y^2 = x^3 + 2x$ with $x,y \in \mathbb{Z}/3\mathbb{Z}$ . (The fourth element of $E(\mathbb{Z}/3\mathbb{Z})$ is the point at infinity.)
   2. Determine the group structure of the group $E(\mathbb{Z}/3\mathbb{Z})$ of order $4$ . [Hint: It is either cyclic or the Klein four group - which one is it?]