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.

    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}$ .)

  3. 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.

  4. 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.

  5. 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?]



William 2007-05-16