Arithmetic, algebra, number theory, sequence and series, analysis, ...
evinda
Posts: 2
Joined: Tue Oct 14, 2014 2:33 pm

Hello!!!!

I want to show that 1/2 is an integer p-adic and to find the first five terms of its powerseries.

Firstly, how do we conclude that 1/2 is an integer p-adic number?

Pengolodh
Posts: 6
Joined: Wed Nov 21, 2012 9:03 am

Well, it isn't an integer for $p=2$. For an odd prime $p$, $1/2=\sum^\infty_{i=0}a_i\,p^i,$ where $a_0=(p+1)/2$ and $a_i=(p-1)/2$ for $i\ge 1$. Just use the geometric series for $1/(1-p)$, multiply by $(p-1)/2$ and add 1.

evinda
Posts: 2
Joined: Tue Oct 14, 2014 2:33 pm

Pengolodh wrote:Well, it isn't an integer for $p=2$. For an odd prime $p$, $1/2=\sum^\infty_{i=0}a_i\,p^i,$ where $a_0=(p+1)/2$ and $a_i=(p-1)/2$ for $i\ge 1$. Just use the geometric series for $1/(1-p)$, multiply by $(p-1)/2$ and add 1.
I want to show that $\frac{1}{2}$ is an integer $5$-adic. How can I do this?
$1/2=\sum^\infty_{i=0}a_i\,p^i,$ where $a_0=(p+1)/2$ and $a_i=(p-1)/2$ for $i\ge 1$