Wilson Primes Proposal of Asymptotic Approximation in the Case of Infinite Existence and Polynomial Approach via Root Multiplicity

Wilson primes are a specific case of a set of primes with a rather particular complexity, resulting froma factorial that forms part of the structural basis of their definition. This factorial relation brings a highcomputational cost for analysis and mapping, creating a limitation on progress beyond a certain threshold. Theaim of this work is to better understand these relations and to propose two approaches. The first concerns anapproximation function considering the existence of more terms in this set, making use of a remarkablemathematical tool proposed by Stirling; and the second shifts the perspective from modular analysis to a higherlevelview, through polynomial analysis, with the focus on finding a connection with the modular form that givesthe group its name. Finally, conclusions, results, and discussions for future work are presented.