The Congruent Number Problem and its Variants

Mentor: Evan Warner

We call a positive integer "congruent" if it is the area of some right triangle with all side lengths equal to rational numbers. For example, 6 is a congruent number, as it is the area of the right triangle with side lengths 3, 4, and 5. The problem of determining which numbers are congruent has a distinguished history (reaching back at least a millennium, and arguably all the way back to Diophantus in the 3rd century AD), yet is still partially unresolved. The basic strategy is to translate the problem into a question about finding rational points on certain explicit curves. The congruent number problem itself has been much-studied, but there are several variants that present more amenable questions (and also have corresponding translations into finding points on curves). For example, what can we say if we replace a right angle with another fixed angle? What if we allow the side lengths to have values in some number field other than the rationals? Depending on the inclination of the participants, this project could involve a substantial computational aspect, probably using SAGE.