Abstract
The Pythagorean equation is extended to higher dimensions via circulant matrices. This form allows for the set of solutions to be expressed in a clean yet non-trivial way. The cubic case, namely the equation x3+y3+z3−3xyz=1, was studied by Ramanujan and displays many interesting properties. The general case highlights the use of circulant matrices and systems of differential equations. The structure of the solutions also allows parametrized solutions of the Fermat equation in degrees 3 and 4 to be given in terms of theta functions.
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