The Congruent Number Problem

A positive integer n is called a ``congruent number'' if it is the area of a right triangle with all rational sides. That is, n is congruent if and only if we can solve the following two equations

$$x^2+y^2=z^2 \hbox{ and } n= {1\over 2} xy$$

for rational numbers x, y, and z. For example, if n=6 then we can choose x=3, y=4, and z=5. The problem is to determine all congruent numbers n. Using this definition, this can be very difficult. The first congruent number is n=5, but the corresponding right triangle does not have integral sides. Generally speaking, the rational numbers occurring for a given congruent number may be extremely complicated. Recently, a simple criterion, that is, simple enough to calculate readily albeit too complicated to be reproduced here, has been discovered that conjecturally determines all congruent numbers.