An Analytical Study of Diophantine Equations of Pythagorean Form: Causal Inferences on Hypothesized Relations between Quadratic and Non-quadratic Triples

In XVII century, presumably between 1637 and 1638, with a note in the margin of Diophantus’ “Arithmetica”, Pierre de Fermat stated that Diophantine equations of the Pythagorean form, , have no integer solutions for , and . Of this statement, however, Fermat never provided a proof. Only after more than 350 years, in 1994, Prof. Andrew J. Wiles was finally successful in demonstrating it (Wiles, 1995; Taylor & Wiles, 1995; Boston, 2008). However, Wiles’ proof adopts calculus techniques far beyond Fermat’s knowledge. Our aim is to show an analytical method to attempt a proof to Fermat’s last theorem with the only use of elementary calculus techniques.