Math 346/646 - Homework 1
Assigned: 8/27.
Due: 9/3 at the start of class.
Notation: Exercise a.b stands for exercise b in Chapter a of Silverman and Tate. Note that the
exercises are all together at the end of the chapter. A problem marked with a ∗ is a challenge
problem.
Problems:
1. Turn in the questionnaire (posted on the class diary) to Jeremy’s office sometime before
class on Wednesday.
2. (a) Show that the only integer solutions to w2 = x2 − x + 1 are those with x = 0 and x = 1.
(Hint: Write the equation as 4w2 − (4x2 − 4x + 1) = 3.)
(b) Find, with proof, all integer solutions to the equation y 2 = x3 − x2 + x.
3. ∗ Prove that there are infinitely many integer solutions to x2 − 5y 2 = 1. Here are the first
few solutions:
x
y
1
0
9
±4
161
±72
2889
±1292
51841 ±23184
930249 ±416020
Based on these examples, try to figure out how to generate the next solution from the previous
one, and use induction to prove that there are infinitely many.
1