Related papers: Fermat's Last Theorem for the Exponent 3

Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat's Last Theorem over $\mathbb Q(i)$. Under the same assumption, we also prove that, for all prime exponents $p \geq 5$, Fermat's equation $a^p+b^p+c^p=0$…

Let $K$ be a number field and $p$ a prime number $\geq 5$. Let us denote by $\mu_p$ the group of the $p$th roots of unity. We define $p$ to be $K$-regular if $p$ does not divide the class number of the field $K(\mu_p)$. Under the assumption…

Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set…

In this article, I discuss material which is related to the recent proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of…

Even though flt is a number theoretic result we prove that the result depends on the topological as well as the field structure of the underlying space.

We study the Generalized Fermat Equation $x^2 + y^3 = z^p$, to be solved in coprime integers, where $p \ge 7$ is prime. Using modularity and level lowering techniques, the problem can be reduced to the determination of the sets of rational…

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…

Fermat's Last theorem (FLT) famously states that the equation $x^n+y^n=z^n$ has no solution in positive integers $x, y, z$ for any integer exponent $n>2$. But does this theorem have a quantitative version? Upon initial investigation we…

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…

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…

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,…

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 $2p+1$…

In a recent paper, the first author provided some lower bounds to solutions of the equations of Fermat and Catalan, based on local power series developments at the ramified prime of a prime cyclotomic extension. Although both equations have…

In this short article we do not prove Fermat's last theorem. We show that the number 2 is an exceptional number in this theorem.

This paper proposes a generalized ABC conjecture and assuming its validity settles a generalized version of Fermats last theorem.

Fermat Last Theorem, which inspired mathematicians during 300 years, is proved by Andrew Wiles. Even among mathematicians there is a narrow circle of specialists, who can read this proof and understand all details. Is it a reason for…

The ABC conjecture implies many conjectures and theorems in number theory, including the celebrated Fermat's Last Theorem. Mason-Stothers Theorem is a function field analogue of the ABC conjecture that admits a much more elementary proof…

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 give a proof of Fermat's little theorem which does not use nor arithmetic(Euclidean algorithm) neither algebra (group theory), but it rather employs the field of the formal power series Q((x)). The note is an example of a mathematical…

In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working…