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

摘要

This paper develops a framework of algebra wherebyevery Diophantine equation is made quickly accessible by a study of thecorresponding row entries in an array of numbers which we call the Binomialtriangle. We then apply the framework to the discussion of some notable resultsin the theory of numbers. Among other results, we prove a new and completegeneration of all Pythagorean triples (without necessarily resorting totheir production by examples), convert the collection of Binomial triangles toa Noetherian ring (whose identity element is found to be the well-known Pascaltriangle) and develop an easy understanding of the original Fermat’sLast Theorem (FLT). The application includes the computation of theGalois groups of those polynomials coming from our outlook on FLT and anapproach to the explicit realization of arithmetic groups of curves by atreatment of some Diophantine curves.