Furthermore, since the greatest common divisor of $\displaystyle p$ and $\displaystyle \theta-1$ is $\displaystyle 1$, we can write $\displaystyle 1=ap+b(\theta-1)$ for some integers $\displaystyle a$ and $\displaystyle b$.

- Show that every integer $\displaystyle x$ can be written in the form $\displaystyle x=y^p+j\theta$ for some integer $\displaystyle y$ and some integer $\displaystyle j$.

- Prove the First Case of Fermat's Last Theorem for the exponents $\displaystyle 13,17$ and $\displaystyle 19$. (That is, show that if there is a nonzero integer solution to $\displaystyle x^p+y^p=z^p$ for $\displaystyle p=13$ then $\displaystyle 13$ is a factor of $\displaystyle xyz$. and so on.)

Any help would be highly appreciated.

Thank you.