We show by explicit construction that Euler's Diophantine equation a4+b4+c4+d4=e4 has infinitely many integral solutions with all terms non-zero. While the arithmetic geometry of elliptic curves provides a context and mathematical description of the methods used, these methods involve mostly elementary algebra. All calculations are carried out explicitly, and any reader could easily use them to find additional new solutions. Euler's equation is specialized to a4+b4+c4+c4 = (a+b+c+d)4 for which one solution is already known. This equation is rewritten into a pair of equations that describe a family of elliptic curves, and we show that the known solution is a non-torsion point on its member of the family. That is to say that the known solution is a point of infinite order in the Abelian group formed on its particular elliptic curve. This tells us that the equation has infinitely many solutions. While this construction can be used to find new Euler solutions, those it produces are very large. To find solutions of more moderate size, we add two additional variables. These allow us to construct an equation in 6 homogeneous variables that describes a higher dimensional surface covering the surface given by the specialized Euler equation. We give an explicit formula for transforming solutions to the new equation into solutions of Euler's equation. The new equation is such that any one solution easily leads to many others. The construction provides any number of ways of changing one solution into another. Thus it is possible to add more examples to previously known 88 solutions to Euler's equation. The paper ends with a list of new solutions to Euler's equation obtained this way.
Document Type: Research Article
Publication date: March 1, 2008
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