Reading

Read pages 1-15 of Silverman-Tate. Try each of the following problems, but be able to present a solution to the one with your name next to it:

1. (Jeff) Prove that the line connecting two distinct rational points in the plane is defined by an equation $ax+by+c$ with $a,b,c\in\mathbf{Z}$, then prove that the intersection of any two distinct rational lines in the plane is empty or a single rational point.

2. (Jennifer) Find all right triangles with integer side lengths and hypotenuse $<30$.

3. (Mauro) For each of the following conics, either find five rational points or prove that there are no rational points:
   1. $x^2 +y^2=6$
   2. $3x^2+5y^2=4$
   3. $3x^2+6y^2=4$

4. (Alex) Draw a rough graph of the conic $x^2 - y^2 = 1$, then give a formula for all the rational points on this conic.

5. (Jenna) Use induction on $n$ to prove that for every $n\geq 1$, the congruence
   
   $$x^2 + 1 \equiv 0 \pmod{5^n}$$
   
   
   has a solution $x_n \in \mathbf{Z}/5^n\mathbf{Z}$.