Chapter 2, Back to the Beginning Video Solutions, Modern Cryptography and Elliptic Curves: A Beginner’s Guide

Problem 1

Analogously to what we did above, find a parametrization for the points on the circle $x^2+y^2=2$ and extract a characterization of the rational points.
To start, project from the rational point $(1,1)$. Note: projecting onto the $x$ or $y$-axis does not work as expected, as not all lines from $(1,1)$ to points on the circle intersect those axes. Instead, try to project onto the line $y=-x$.

Sriparna Bhattacharjee

Numerade Educator

Problem 2

Now consider the issue of rational points on $x^2+y^2=3$. In contrast to the examples above, prove that there are no rational points on this curve, and describe the crucial difference between this example and the one before.

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01:47

Problem 3

Find a square-free congruent number not in the list above, and show all the work to obtain it.

Gregory Higby

Numerade Educator

01:09

Problem 4

Find all of the rational points on the curve $x^n+y^n=1$ where $n$ is an integer, $n>2$.

Carson Merrill

Numerade Educator

Problem 5

Let $V$ be the set of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy the differential equation $f^{\prime \prime}-f=0$. Show that $V$ is a vector space over $\mathbb{R}$ and, assuming its dimension is 2 , find a basis for $V$.

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03:20

Problem 6

Using the ideas above, prove the Bachet duplication formula for $y^2=x^3+k, k \neq 0$.
$$
(x, y) \mapsto\left(\frac{x^4-8 k x}{4 y^2}, \frac{-x^6-20 k x^3+8 k^2}{8 y^3}\right) .
$$

We outline some useful steps.
(1) Use implicit differentiation to derive a formula for the slope of the tangent line to the curve $y^2=x^3+k$ which is valid at any point $(x, y)$ where $y \neq 0$.
(2) Now write down the equation of the tangent line to the curve at the point $(a, b)$ where we assume $b \neq 0$. It will be convenient if you use $m$ for the slope for the time being until you need to use its actual value.
(3) Now we want to find the point(s) of intersection of the tangent line with the cubic, and this requires a little work. Substitute the expression for $y$ given by the line into the equation that defines the cubic results in an equation of the form $f(x)=0$ where $f$ is a polynomial of degree 3 . Your job is to factor the polynomial since its roots are the $x$-coordinates corresponding to the points of intersection. Here we catch a bit of a break. Certainly one of the roots is $a$, which means $(x-a)$ is a factor. But it should not be too much of a surprise that $a$ is (at least) a double root since the line is tangent to the curve at $x=a$ (much like $y=(x-a)^r$ is tangent to the $x$-axis at $x=a$ and the root $a$ has multiplicity $r$ ). After factoring out the first of the $(x-a)$ factors, it would be a good time to put in the real value of $m$ to see what simplifies.

Stark Ledbetter

Numerade Educator

Problem 7

Properties of rational lines in the plane.
(1) Is every point on a rational line a rational point?
(2) If a line passes through at least two rational points, is it a rational line? What about lines if we only know one rational point through which they pass?
(3) Consider two distinct rational lines which intersect. Do they intersect in a rational point?

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04:47

Problem 8

Characterizing the intersection of lines and conics.
(1) In how many points can two arbitrary lines (in the plane) intersect?
(2) In how many points can a line and a conic intersect?

William Nute

Numerade Educator

06:12

Problem 9

In the questions below, we assume all the plane curves are irreducible, meaning they are the zero sets of polynomials $f(x, y)$ where $f(x, y)$ is irreducible. It follows (from abstract algebra) that two distinct irreducible plane curves can only intersect in a finite number of points. The questions below try to get at discovering what that number might be.
For all the problems below, consider your curves in $\mathbb{R}^2$. Can you come up with examples that suggest answers to these questions? Can you prove any of your assertions?
(1) In how many points can two (distinct) conics intersect?
(2) In how many points can a conic and a cubic intersect?
(3) In how many points can two (distinct) cubics intersect?
(4) What would be your guess for a generalization?
(5) Consider the intersection of a rational line with a rational conic.
(a) Are the point(s) of intersection necessarily rational? Give a proof or provide a counterexample.
(b) Now let's suppose that the line intersects the conic in two points, one of which is rational. Is the second point necessarily rational? Give a proof or a counterexample.

Kevin Maritato

Numerade Educator

01:05

Problem 10

As a simple example show that the curve $y=$ $(x-a)^k$ intersects the $x$-axis with multiplicity $k$ at $x=a$ and with multiplicity 0 at all other points $x=b$.

Carson Merrill

Numerade Educator

10:11

Problem 11

Next, let's gain a little more insight by examining the case of zeroes of order 1 and 2 . Let $h(x)$ be a polynomial of degree $n \geq 2$ with coefficients in a field $F$, and let $a \in F$. Then prove the following.
(1) $h(x)=(x-a) q(x)+h(a)$ for some polynomial $q$ having coefficients in $F$.
(2) $h(a)=0$ if and only if $h(x)=(x-a) q(x)$.
(3) $h$ has a double root at $a$ if and only if $h(a)=h^{\prime}(a)=0$, where $h^{\prime}(x)$ is the first derivative of $h(x)$.

Anthony Ramos

Numerade Educator

10:11

Problem 12

Establish the following generalization of the work we have started above. Show that $h$ has a zero of order $k$ at $x=a$ if and only if $h(a)=h^{\prime}(a)=\cdots=h^{(k-1)}(a)=0$ and $h^{(k)}(a) \neq 0$, where $h^{(i)}$ is the ith derivative of $h$. Hint: Taylor polynomials are your friend.

Anthony Ramos

Numerade Educator

01:15

Problem 13

Now consider $y^2=g(x)$ where $g$ is a cubic, that is the zero set of $f(x, y)=y^2-g(x)$. We want to see that a nonvertical tangent has multiplicity at least 2 at the point of tangency.

Carson Merrill

Numerade Educator