In an anonymous Arab manuscript from the year 927, the following question is posed: given a squarefree integer $$n$$, does there exist a rational number $$a$$ such that $$a-n, a, a+n$$ are all squares of rational numbers? The examples 5 and 6 were known to the Arabs, and later Fibonacci discovered that 7 was a congruent number. In the other direction, Fermat proved that 1 was not congruent -- this is equivalent to his famous Last Theorem in the special case that the exponent is 4. However, to this day, there is no algorithm for this problem in general. Many advances on this problem were made using the theory of elliptic curves and, under the assumption that the Birch and Swinnerton-Dyer conjecture is true (a current Millenium Prize Problem), there is a fast algorithm to determine whether a given integer is congruent. This class will first attack the congruent number problem using elementary methods, and then discuss how elliptic curves come in.