As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

Please follow this blog

Search this blog

Sunday, December 21, 2014

Dirichlet ' s Theorem

$$p / ( p-1 ) ! + \left(\frac{a}{p}\right)  a^{(p-1)/2}$$

Proof.
Consider the equation $Ax \equiv a \bmod p$ with $A, x \in \{ 1,2, \cdots p-1\}$,

Case $\left(\frac{a}{p}\right)=-1$
In this case $x$ and $A$ are different members of the set $\{ 1,2, \cdots p-1\}$, there are $(p-1)/2$ distinct pairs $(A, x) $ and pairwise multiplication gives the following identity: $( p-1 ) ! = a^{(p-1)/2}$.

Case $\left(\frac{a}{p}\right)=1$
In this case $a$ is a quadratic residue of $p$ so there are two pairs where $x$ and $A$ are equal members of the set $\{ 1,2, \cdots p-1\}$, there are $(p-3)/2$ distinct pairs $(A, x) $ and pairwise multiplication gives the following identity: $\frac {( p-1 ) ! }{k (p-k)}= a^{(p-3)/2}$.
Now $k( p-k) = kp - k^2 \equiv -a \bmod p $.  Another pairwise multiplication gives the following identity: $( p-1 ) ! = - a^{(p-1)/2}$.

Combining both cases and replacing the sign with the Legendre symbol gives $$p / ( p-1 ) ! + \left(\frac{a}{p}\right)  a^{(p-1)/2}.$$

Popular Posts

Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before




(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)