Platonic Solids and Solutions to X2 + Y3 = dZr

Abstract

This manuscript investigates the properties of the diophantine equation

X2 + Yr

Here d is a given integer, r is one of 3,4, or 5 and the unknowns X,Y ,Z are required to be integers with no common factor other than ±1.

The equation is a special case of the so called generalized Fermat equations. The equation is classified as a spherical generalized Fermat equation. As such, it has many properties in common with the better known (and in fact studied by the Babylonians since at least 1600 BC) Pythagoras equation X2 + Y2 = Z2.

There is an infinite number of solutions to the Pythagoras equation. Many will remember the triples (3, 4, 5) and (5, 12, 13) from their high school days. An infinite set of solutions can be obtained by assigning integer values to (s, t) in the formula X = s2-t2, Y = 2st, Z = s2+t2. In fact, all solutions to the Pythagoras equation can be obtained if we also allow X to be swapped with Y, and Z to be replaced by -Z.

Frits Beukers showed in 1998 that similar thing happens withX2 + Y3 = dZr. There is an infinite number of solutions and these solutions can all be obtained by assigning integer values to (s, t) in a finite set of formulae for (X, Y, Z). The main problem is then to find these formulae (parameterizations). Although in principle Beukers’ method is effective, it seems difficult to obtain all the necessary parameterizations by using it directly.

This manuscript describes an algorithm to generate complete sets of parameterizations to these equations. In particular, it gives a complete set of parameterizations to the hitherto inaccessible equation X2 + Y3= Z5 (there are 27 parameterizations to this equation, or 49 if solutions with X and -X are considered distinct).

The method lends on several mathematical techniques from Invariant Theory - an important branch of mathematics at the end of the 19th century. The title of the thesis is explained by the fact that the 60 symmetries of the 20-sided platonic solid called the icosahedron play a key role in producing solutions to the equation X2 + Y3 = dZ5.