On a Possible Generalization of Fermats Last Theorem

In the present paper we study, in a mathematically non-formal way, the validity of the Fermat's Last Theorem (FLT) by generalizing the usual procedure of extracting the square root of non convenient objects initially introduced by P. A. M. Dirac in the study of the linear relativistic wave equation.

We give again the proof of several classical results concerning the cyclotomic approach to Fermat's last theorem using exclusively class field theory (essentially the reflection theorems), without any calculations. The fact that this is possible suggests a part of the logical inefficiency of the historical investigations. We analyze the significance of the numerous computations of the literature, to show how they are probably too local to get any proof of the theorem. However...

The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f(x) in Z[x] monic and q1 ,...., qm in Z, we study the conditions for which the Diophantine equatio (y + q1 )(y + q2 ) .... (y + qm ) = f(x) has finitely many solutions in integers. Also assuming ABC Conjecture, we study the conditions for finiteness of integer solutions of the Diophantine equation f(x) = g(y).

This is a continuation of my work on Arithmetic Teichmuller Spaces developed in the present series of papers. In this paper, I show that the Theory of Arithmetic Teichmuller Spaces leads, using Shinichi Mochizuki's rubric, to a proof of the $abc$-conjecture (as asserted by Mochizuki).

With a simple transformation of the three exponents the generalized Fermat equation can be put into the same form as the Fermat equation. When it is rewritten into this new altered form any real solutions to the altered equation equal a subset of real solutions to the Fermat equation. This result, along with a paradigm previously established by G. Frey and Y. Hellegouarch, can be used to show that the generalized Fermat equation has no solutions in coprime integers. In fact w...

One shows that the Last Fermat Theorem is equivalent to the statement that all rational solutions of the famous equation are provided by an orbit of rationally parametrized subgroup of a group preserving k-ubic form. This very group naturally arrises in the generalized Clifford algebras setting .

We consider a variant of the ABC Conjecture, attempting to count the number of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC Conjecture is equivalent to the statement that for $a+b+c<1$, the number of solutions is bounded independently of $N$. If $a+b+c \geq 1$, it is conjectured that the number of solutions is asymptotically $N^{a+b+c-1 \pm \epsilon}.$ We prove this conject...

Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would follow if it were known for abc sums that are not highly divisible. We prove both theorems for general number fields.

We formulate an exponential Diophantine equation, which is is some sense one order higher that Fermat's Last Theorem. We also give three examples of solutions to this exponential Diophantine equation and formulate a conjecture.

In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c\,\log X$ prime numbers $p \leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by B\"ockle-Gu...