Ms Amy Batsiokis
The Congruent Number Probem
The congruent number problem was first discussed in the tenth century by Arabic
scholars and it is still unsolved today. At present, we have an almost complete
solution to the problem (but it relies on the weak form of the Birch and Swinnerton-
Dyer conjecture [in Chapter 8]) being proven true.
This thesis begins with some elementary work on pythagorean triples, which is
followed by two equivalent definitions of congruent numbers. We then look at elliptic
curves which lead to yet another alternate condition for a number to be congruent.
This is followed by a look at some objects which are related to elliptic curves:
height pairing, the L-function, and the Shafarevich-Tate group. Then everything
is tied together in the last chapter which is about the Birch and Swinnerton-Dyer
conjecture, which has not been proved yet, and Tunnelâs Theorem which will provide
a complete answer to the congruent number problem when the weak form of the
Birch and Swinnerton-Dyer conjecture has been proven.