Journal of Applied Mathematics & Bioinformatics

A study of Fermat’s Last Theorem and other Diophantine Equations

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•                                                     Abstract

   

  This paper develops a framework of algebra whereby every Diophantine equation is made quickly accessible by a study of the corresponding row entries in an array of numbers which we call the Binomial triangle. We then apply the framework to the discussion of some notable results in the theory of numbers. Among other results, we prove a new and complete generation of all Pythagorean triples (without necessarily resorting to their production by examples), convert the collection of Binomial triangles to a Noetherian ring (whose identity element is found to be the well-known Pascal triangle) and develop an easy understanding of the original Fermat’s Last Theorem (FLT). The application includes the computation of the Galois groups of those polynomials coming from our outlook on FLT and an approach to the explicit realization of arithmetic groups of curves by a treatment of some Diophantine curves.