Wilson’s Theorem states that if p is a prime number, then the product of the first (ip /i- 1) positive integers, increased by one is divisible by p. This classical result in number theory was attributed to John Wilson by Edward Waring in 1770. The first known proof of Wilson’s Theorem was published in 1771 by the French mathematician Joseph-Louis Lagrange. Wilson’s Theorem has applications in primality testing, cryptography, and various other areas of mathematics. In this article, we shall prove that if ip/i is a prime number, then for each natural number,i j/i, that lies between 1 and (ip /i- 2), the binomial coefficient (ip /i- 1) choose ij/i, decreased by the ij/ith power of negative one, is divisible by p. As such, these new results can be viewed as extensions of Wilson’s Theorem. We shall also prove that if ip/i is a prime number, then the square of the factorial of the ratio of (ip /i- 1) and two, is congruent to either 1 modulo ip/i or - 1 modulo ip/i. Additionally, we shall prove that for all positive integersi m/i bigger than or equal to 5, im/i is prime if and only if im/i divides the sum of either one or negative one and the square of the factorial of the ratio of (im /i- 1) and two. Next, we shall use some of the schemes developed earlier to investigate the behavior of the divisors of the sum of prime powers of relatively prime positive integers ia, b/i. Lastly, we shall show that if ip/i is a prime number and ik/i is a positive integer, then the ratio of the sum of prime powers of ia, b/i and the sum of ia, b/i is not divisible by the ik/ith power of ip/i.
Joseph Gaskin (Fri,) studied this question.