Fermat's Theorem

• Feb 16th 2013, 03:44 PM
Gibo
Fermat's Theorem
Hi,

I have difficulties with below problem:

Let $\displaystyle p$ and $\displaystyle \theta$ be primes with $\displaystyle \theta>p$ such that $\displaystyle p$ is not a factor of $\displaystyle \theta - 1$. As $\displaystyle \theta$ is a prime, we know that for any$\displaystyle x\epsilon Z$ we can write $\displaystyle x^{\theta-1}$ in the form$\displaystyle x^{\theta-1} = k\theta+1$ for some integer $\displaystyle k$. 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.
• Feb 16th 2013, 04:06 PM
HallsofIvy
Re: Fermat's Theorem
• Feb 16th 2013, 04:12 PM
HallsofIvy
Re: Fermat's Theorem
Quote:

Originally Posted by Gibo
Hi,

I have difficulties with below problem:

Let $\displaystyle p$ and $\displaystyle \theta$ be primes with $\displaystyle \theta>p$ such that $\displaystyle p$ is not a factor of $\displaystyle \theta - 1$. As $\displaystyle \theta$ is a prime, we know that for any$\displaystyle x\epsilon Z$ we can write $\displaystyle x^{\theta-1}$ in the form$\displaystyle x^{\theta-1} = k\theta+1$ for some integer $\displaystyle k$.

Are your sure we know this? If, for example, $\displaystyle \theta= 3$ and $\displaystyle x= 6$, this says that $\displaystyle 6^{3-1}= 6^2= 36= 3k+ 1$. And I don't believe this is true.

Quote:

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.
• Feb 16th 2013, 11:41 PM
Gibo
Re: Fermat's Theorem
Thank you very much for pointing that, I found it really helpful.