Date of Graduation
Master of Science in Mathematics
equilateral triangle, parametric family of solutions, dehomogenize, singular point, tangent cone, cuboctahedron
This thesis explores some techniques for obtaining integer solutions to the Diophantine equation a^2b^2+a^2c^2+a^2d^2+b^2c^2+b^2d^2+c^2d^2=a^4+b^4+c^4+d^4. We show that, equivalently, we can find rational solutions to the related equation x^2y^2+x^2z^2+y^2z^2+x^2+y^2+z^2=x^4+y^4+z^4 which we find by dehomogenizing our original homogeneous equation. Methods are provided for generating infinitely many parametric families of rational solutions to this equation, starting with just a single rational point on the surface of f (x, y, z). These techniques are applied to two cases which we explore in depth: the "degenerate” case where the point in question lies on one side of an equilateral triangle, and the case where three of the values in our original equation are in arithmetic progression. We examine the singular points of the surface of f (x, y, z) and use them to generate even more solutions to our original equation. Finally, we consider the planes containing at least three singular points of the surface and realize that these give rise to the degenerate case and the case in which three terms are in arithmetic progression. Ultimately, we find that although the exploration of these two cases may have seemed intended only to demonstrate techniques for generating solutions, they would in fact have been forced with the consideration of the singular points of the surface.
© Christina P. Bisges
Bisges, Christina P., "Geometric Techniques for Solving a Certain Diophantine Equation" (2012). MSU Graduate Theses. 1642.