Topology of Numbers · Problem 2

The Farey Diagram — Allen Hatcher

Mathematics — Number Theory · difference-of-squares

Problem

(a) Find all the positive integer solutions of

x^2 - y^2 = 512

by factoring

x^2 - y^2

(x+y)(x-y)

and considering the possible factorizations of

512

. (b) Show that the equation

x^2 - y^2 = n

has only a finite number of integer solutions for each value of

n > 0

. (c) Find a value of

n > 0

for which the equation

x^2 - y^2 = n

has at least 100 different positive integer solutions.