Archives of Interdisciplinary Education
Solving Fermat Last Theorem and the Generalized Diophantine Equations
Abstract
Nikos Mantzakouras and Carlos Lopez Zapata
The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics; a simple relation of square numbers that encapsulates the glory of mathematical science. It is also justifiably the most popular but also the most sublime theorem in mathematical science. The starting point was Diophantus’ 20th problem (Book VI of the Arithmetica of Diophantus use here is based on elementary inequalities and gives a solution to any Diophantine equation of degree n with respect to the number of variables d. It is a method that does not exclude other equivalent analytic methods for finding specific solutions. However, the proposed method generalizes to any exponential positive integer n ∈ Z + ≥ 3, with number of variables d≥3. An explicit proof is also given for variables in both parts with equal or different numbers of variables, also of degree n. By this logic we can obtain many extensions to other diophantine equations that present great difficulty in being treated by diophantine analysis and number theory. As a direct application it is Fermat’s Last Theorem and of course hasmuch broader generalization to symmetric or asymmetric Diophantine equations.