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Given 
$k\geqslant 2$ , we show that there are at most finitely many rational numbers 
$x$ and 
$y\neq 0$ and integers 
$\ell \geqslant 2$ (with 
$(k,\ell )\neq (2,2)$ ) for which In particular, if we assume that 
$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$
$\ell$ is prime, then all such triples 
$(x,y,\ell )$ satisfy either 
$y=0$ or 
$\ell <\exp (3^{k})$ .
Let 
$K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over
$K$ we mean the statement that there is a constant 
$B_{K}$ such that for any prime exponent 
$p>B_{K}$, the only solutions to the Fermat equation are the trivial ones satisfying 
$$\begin{eqnarray}a^{p}+b^{p}+c^{p}=0,\quad a,b,c\in K\end{eqnarray}$$
$abc=0$. With the help of modularity, level lowering and image-of-inertia comparisons, we give an algorithmically testable criterion which, if satisfied by 
$K$, implies the asymptotic Fermat’s Last Theorem over 
$K$. Using techniques from analytic number theory, we show that our criterion is satisfied by 
$K=\mathbb{Q}(\sqrt{d})$ for a subset of 
$d\geqslant 2$ having density 
${\textstyle \frac{5}{6}}$ among the squarefree positive integers. We can improve this density to 
$1$ if we assume a standard ‘Eichler–Shimura’ conjecture.
We discuss a clean level lowering theorem modulo prime powers for weight 2 cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations 
$a{{x}^{n}}\,+\,b{{y}^{n\,}}\,+\,c{{z}^{n}}\,=\,0$.
