Related papers: Fermat's Last Theorem over ${\mathbb Q}(\sqrt{5})$…
The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain has obtained an important result in this direction by establishing that, if $2p+1$…
We formalise the proof of the first case of Fermat's Last Theorem for regular primes using the \emph{Lean} theorem prover and its mathematical library \emph{mathlib}. This is an important 19th century result that motivated the development…
This article deals with a conjecture, introduced in [GQ] (hereinafter $SFLT2$), which generalizes the second case of Fermat's Last Theorem: {\it Let $p>3$ be a prime. The diophantine equation $\frac{u^p+v^p}{u+v}=w_1^p$ with $u,v,u+v,…
Following the famous proof of Fermat's Last Theorem by Andrew Wiles using the modularity of elliptic curves over $\mathbb{Q}$, significant developments have been made in the study of Diophantine equations using the modularity method. This…
Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this…
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 $Ax^p +By^p +Cz^p = 0$,…
Let $p \ge 5$ be a prime and let $\mathbb{Q}_{n,p}$ denote the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. We show that $\mathbb{Q}_{n,p}$ has no exceptional units. We use this to prove the effective asymptotic…
We study Fermat's Last Theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions (B,e) of models of arithmetical theories (in the language L=(0,1,+,x,<)) by a binary (partial or total)…
This article, complement to the article [Que], deals with some generalizations of Futw\"angler's theorems for the second case of Fermat's Last Theorem (FLT2). Let $p$ be an odd prime, $\zeta$ a $p$th primitive root of unity, $K:=\Q(\zeta)$…
We propose a new approach at Fermat's Last Theorem (FLT) solution: for each FLT equation we associate a polynomial of the same degree. The study of the roots of the polynomial allows us to investigate the FLT validity. This technique,…
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.…
We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of $a+b=c$ implies that there exist infinitely many primes. In particular, for small exponents such as $n=3$ or $4$ this gives a new proof of…
We show that an elementary proof of Fermat's Last Theorem (FLT) exists. Our paper also extends the scope of FLT from integers to all rational numbers.
We present an elementary proof of Fermat's Last Theorem. No ancillary results are used, not even the most basic ones. The proof directly leads to a contradiction of the Fermat equation in the set of integers.
This research focuses on the Numerical approach for Fermat's Last theorem. We can induce an Alternative form of Fermat's last theorem by using particular geometric mapping $\mathcal{M}$ on a Cartesian plane to a Torus. It transforms the…
Fermat's Last Theorem is proved by using the philosophical and mathematical knowledge of 1637 when the French mathematician Pierre de Fermat claimed to have a truly marvelous proof of his conjecture. Our approach consists of setting three…
We prove, in the context of the section conjecture, that every Selmer section over $\mathbb{Q}$ of the affine Fermat curve with exponent $\ell$ is cuspidal for $\ell\geq 7$.
Wiles' proof of Fermat's last theorem initiated a powerful new approach towards the resolution of certain Diophantine equations over $\mathbb{Q}$. Numerous novel obstacles arise when extending this approach to the resolution of Diophantine…
The divisibility restrictions in the famous equation a n+bn=cn in Fermat Last Theorem (FLT, 1637) is analyzed how it selects out many triples to be Fermat triple (i.e. solutions) if n greater than 2, decreasing the cardinality of Fermat…
Let $q > 5$ be a prime and $K$ a quadratic number field. In this article we extend a previous result of Najman and the author and prove that if $E/K$ is an elliptic curve with potentially multiplicative reduction at all primes $\mathfrak q…