Topology of Numbers · Problem 14

The Farey Diagram — Allen Hatcher

Mathematics — Number Theory · rational-points-fermat-curve

Problem

Show that if the curve

x^n + y^n = 1

has a rational point with

x

and

y

nonzero, then it has a rational point with

x

and

y

positive. Hint: Consider the equation

a^n + b^n = c^n