On a Possible Generalization of Fermats Last Theorem

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 Newtonian triangles. We then apply this framework to the understandimg of the Fermat's Last Theorem and discuss some of its direct consequences.

This paper is submitted to Algebraic-Number-Theory Archives for validation by Number Theorists Community. It is an update of the previous versions ANT-0155, ANT-0170, ANT-0205, ANT-0237, ANT-0321, ANT-0333, and ANT-0356, of which the first four were titled `A generalization of Eichler criterium for Fermat's Last Theorem' and the last three were titled `A classical approach on Fermat-Wiles theorem'. This version contains a complete reorganization of the paper with a first pa...

While currently the $abc$ conjecture and work towards it remains open or is disputed, at the same time much work has been done on weaker versions, as well as on its generalisation to number fields. Given integers satisfying $a+b=c$, Stewart and Yu were able to give an exponential bound for $\max(a,\,b,\,c)$ in terms of the radical over the integers, while Gy\"{o}ry was able to give an exponential bound in the algebraic number field case for the projective height $H_{K}(a,\,b,...

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat's Last Theorem holds for many totally real fields. Later this result was extended by Deconinck to generalized Fermat equations of the form $Ax^p +By^p +Cz^p = 0$, where A;B;C are odd integers belonging to a totally real field. Another extension was given by Sengun and Siksek. They showed that the Fermat equation holds asymptotically for imaginary quadratic number fields satisfying usual conjectures...

This note imparts heuristic arguments and theorectical evidences that contradict the abc conjecture over the rational numbers. In addition, the rudimentary datails for transforming this problem into the doimain of equidistribution theory are provided.

Let $K$ be a totally real number field, and $ \mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation, i.e., $Ax^p+By^p+Cz^p=0$ over $K$ of prime exponent $p$, where $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even (in the sense that $\mathfrak{P}| ABC$, for some prime ideal $\mathfrak{P}$ of $ \mathcal{O}_K$ with $\mathfrak{P} |2$). For certain class of fields $K$, we prove that the equa...

This paper presents a novel direct elementary proof for Fermat's Last Theorem. We use algebra, modular math, and binomial series to develop inherent mathematical relationships hidden within Fermat's Last Theorem. With these derived relationships, we are able to develop general pattern applicable for all positive integers of n. Finally, we are able to confirm and complete the direct proof for Fermat's Diophantine equation for all n.

This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than a fixed power p of log H. Assuming the abc Conjecture we show that there are finitely many solutions if p<1. We discuss a conditional result, showing that the Generalized Rieman...

The abc conjecture is one of the most famous unsolved problems in number theory. The conjecture claims for each real $\epsilon > 0$ that there are only a finite number of coprime positive integer solutions to the equation $a+b = c$ with $c > (rad(a b c))^{1+\epsilon}$. If true, the abc conjecture would imply many other famous theorems and conjectures as corollaries. In this paper, we discuss the abc conjecture and find new applications to powerful numbers, which are integers ...

Fermat's Last Theorem (FLT) implies that the Frey curves do not exist. A proof of FLT independent of proved Taniyama-Shimura-Weil conjecture is presented.