An Interesting Property of Certain Binomial Coefficients That Leads to a Fascinating Refinement of Wilson’s Theorem

Wilson’s Theorem states that if p is a prime number, then the product of the first (<i>p </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 <i>p</i> is a prime number, then for each natural number,<i> j</i>, that lies between 1 and (<i>p </i>- 2), the binomial coefficient (<i>p </i>- 1) choose <i>j</i>, decreased by the <i>j</i>th 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 <i>p</i> is a prime number, then the square of the factorial of the ratio of (<i>p </i>- 1) and two, is congruent to either 1 modulo <i>p</i> or - 1 modulo <i>p</i>. Additionally, we shall prove that for all positive integers<i> m</i> bigger than or equal to 5, <i>m</i> is prime if and only if <i>m</i> divides the sum of either one or negative one and the square of the factorial of the ratio of (<i>m </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 <i>a, b</i>. Lastly, we shall show that if <i>p</i> is a prime number and <i>k</i> is a positive integer, then the ratio of the sum of prime powers of <i>a, b</i> and the sum of <i>a, b</i> is not divisible by the <i>k</i>th power of <i>p</i>.