I am so exited to learn math from this site. I posted the question today and I got good replies from members today itself. I will try to answer other number Theory questions in near future. With same confidence and motivation, I am sending TWO more questions to the members. These are also my observations and it may be done by simple proofs.
1) for a positive odd integer $p$, and $p_1$, $p_2$ - two different odd primes, and $p_1+p_2 - p = 1$ then $(p - p_1)!(p - p_2)! = -1 (\mod p)$ iff $p$ is prime.
2) For a prime $p > 7$, $(p,p +2)$ are said to be twin pair primes iff $4(p-3)! + (p + 2)$ divides $p$.
Please justify the above statements.