Question

If ${\mathit{x}}^{\mathbf{k}}\mathbf{+}{\mathit{y}}^{\mathbf{k}}\mathbf{=}{\mathit{z}}^{\mathbf{k}}$has no nonzero integer solutions and$\mathit{k}\mathbf{|}\mathit{n}$ , then show that ${\mathit{x}}^{\mathbf{n}}\mathbf{+}{\mathit{y}}^{\mathbf{n}}\mathbf{=}{\mathit{z}}^{\mathbf{n}}$ has no nonzero integer solution.

Short answer

It has been proved that$x^n+y^n=z^n$ has no nonzero integer solution.

Step-by-step solution

Given that

Given that$x^k+y^k=z^k$ has no nonzero integer solutions.

Also,$k|n$ .

So let $n=km$.

Prove that xn+yn=zn  has no integer solution

Let$(a,b,c)$ is a solution of$x^n+y^n=z^n$ where $a,b,c\in \mathrm{\mathbb{Z}}$

Then $a^n+b^n=c^n$.

This implies

$\begin{array}{c}{a}^{\left(km\right)}+b^{\left(km\right)}=c^{\left(km\right)}\\ {\left(a^m\right)}^k+{\left(b^m\right)}^k={\left(c^m\right)}^k\end{array}$

This shows that$(a^m,b^m,c^m)$ is a solution of $x^k+y^k=z^k$.

This is contradiction to the given statement.

Hence, $x^n+y^n=z^n$has no integral solution.

Conclusion

It can be concluded that$x^n+y^n=z^n$ has no nonzero integer solution.