After developing a method based on a trigonometric periodic function, to work with prime numbers, I sought to find alternative functions for reasons such as efficiency. One of the most interesting versions found takes the form of a Product Polynomial. This Product Polynomial can be used to show that all primes can be generated by a related family of polynomials, and that the nth prime can always be calculated with a finite polynomial. The panels included here show the form of the product polynomial as well as some of the manipulations of such. The 1st panel shows the form, however the 7 as the index bound was arbitrarily chosen for an example. The 2nd panel shows the relation of the heart of the form to Pochhammer notation. The 3rd panel shows the function expanded and simplified via Wolfram Alpha.
Panels 4 through 11 are mostly random related filler computations, were included chiefly for personal archival purposes, but may still be of interest to some. Panel 12 shows the "Number of Factors Wave" mentioned in my previous prime papers, however here it is being generated with the Product Polynomial.
Lastly, Panel 13 shows a Product Polynomial Primality Test for a number a. Again, the upper bound on the index in this example, 14, is arbitrary. The real requirement on the upper bound is that it be greater than or equal to a. For further understanding on why see the original papers here, https://www.thoughtfarm.info/single-post/2018/01/07/Using-Waves-to-Determine-the-Primes, and here, https://www.thoughtfarm.info/single-post/2018/01/06/A-Series-of-Questions-about-some-Series.