This research offers an extensive analysis of a family of prime-generating polynomials \( P_k(n) = n^2 - (2k - 79)n + [41 + (k - 39)(k - 40)] \), designed to generate prime numbers for integer values of \( n \) up to \( k \) where \( k \in \{1, 2, \ldots, 80\} \). We elaborate on mathematical clarity, consistency in proofs, error bounds, prime repetition analysis, and cryptographic implications. This study provides in-depth inductive proofs, graphical error bounds analysis, prime repetition statistics, and cryptographic security assessments, offering novel insights and future directions for research in number theory and secure prime generation.