A congruent number is defined as a positive integer that equals the area of a rectangular triangle whose sides have rational lengths. The first few congruent numbers are 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, ... The question of whether a given integer is a congruent number turns out to be equivalent to a problem about certain Elliptic Curves. Using congruent numbers as an excuse we will give a gentle introduction to some fundamental problems in Number Theory, such as the Birch Swinnerton-Dyer Conjecture, Fermat's Last Theorem and the Langlands Conjectures.
The talk addresses a general audience with no specific knowledge in Number Theory.