The basic Equations of Number theory are the Diophantine Equations, which are polynomial equations with integer coefficients. One looks for integral, or rational, solutions, and makes use of the structure of real or complex solutions. Simplest non-trivial examples are the Pythagorean and Pell's Equations. Congruent numbers, which are positive integers which arise as areas of rational right triangles, have been understood quite well due to some striking recent results (due to others). At the end of the lecture, we will discuss a generalization to the setting of right rational tetrahedra, which leads to some intriguing geometry.

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