A Note on Fermat Equation＇s Fascination

As former Fermatist, the author tried many times to prove Fermat＇s Last Theorem in an elementary way. Just few insights of the proposed schemes partially passed the peer-reviewing and they motivated the subsequent fruitful collaboration with Prof. Mario De Paz. Among the author＇s failures, there is an unpublished proof emblematic of the FLT＇s charming power for the suggestive circumstances it was formulated. As sometimes happens with similar erroneous attempts, containing out-of-context hints, it provides a germinal approach to power sums yet to be refined.

A Journey into Fermat＇s Equation

As expounded in some recent mathematical conferences, this research on that amazing source of algebraic ideas known as Fermat＇s equation is aimed to prove how Fermat triples can be limited until the impossible existence through a criterion of incompatible parities related to unexplored properties of the binomial coefficients. In this paper, the authors use a technique based on the analysis of four numbers and their internal relations with three basic compulsory factors. It leads to the practical impossibility to find any triple of natural numbers candidate to satisfy Fermat＇s equation, because when the authors try to meet a condition between parity and range the authors are compelled to violate the other one, so that they are irreducibly alternative. In particular, there is a parity violation when the authors choose all the basic factors in the allowed range and the authors obtain exceeding values of one of the involved variables when the authors try to restore the parity. Since Fermat＇s last theorem would consequently be demonstrated, many readers could recall the never found elementary proof of FLT （Fermat＇s last theorem） claimed by Pierre de Fermat. The authors are not encouraging such an interpretation because this paper is intended as a journey into Fermat＇s equation and the reader＇s attitude should be towards the algebraic achievements here proposed, with their possible hidden flaws and future developments, rather than to legendary problems like Fermat＇s riddle.

A Variant of Fermat’s Diophantine Equation

A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.

On Fermat’s Last Theorem and Galaxies of Sequences of Positive Integers

We construct sequences of positive integers which are solutions of the equation x2+y2=z2. We introduce Mouanda’s choice functions which allow us to construct galaxies of sequences of positive integers. We give many examples of galaxies of numbers. We show that the equation x2n+y2n=z2n (n ≥2) has no integer solutions. We prove that the equation xn+yn=zn (n ≥3) has no solutions in N. We introduce the notion of the planetary representation of a galaxy of numbers which allow us to predict the structure, laws of the universe and life in every planet system of every galaxy of the universe. We show that every multiverse contains a finite number of universes.