Question

If $p$ is a prime, show that $|\underset{–}{p-2r}|\underset{–}{2r-1}-1$ is divisible by $p$.

Answer 1

By Wilson's theorem;-

$1+(p-1)!=M\left(p\right)$

$\therefore 1+(p-1)(p-2)........(p-\overline{2r-1})(p-2r)!=M\left(p\right)$

$\therefore 1+\left\{M\right(p)-(2r-1)!\}(p-2r)!=M\left(p\right)$

$1+M\left(p\right)-(p-2r)!(2r-1)!=M\left(p\right)$

$\therefore (p-2r)!(2r-1)!=M\left(p\right)+1$

Hence proved that $(p-2r)!(2r-1)!-1$ is divisible by $p$