Universal Solutions to gcd (n^ (pk) - 1, n! - 1) > 1 via Wagstaff-Type Factors)

This paper provides a formal and constructive solution to the open question #511226 posed on MathOverflow regarding the existence of universal solutions to gcd (n^ (p*k) - 1, n! - 1) > 1 for every odd prime p >= 5. By defining the Wagstaff-type integer Wₚ = (2ᵖ + 1) /3, we unconditionally prove that every prime factor q of Wₚ yields a universal solution given by n = q - 2, satisfying the inequality simultaneously for all integers k >= 1. The proof relies on an elegant combination of a Fermat-type congruence derived from the defining relation q | Wₚ with a direct application of Wilson's theorem. Immediate corollaries guarantee the existence of at least one solution for all odd primes p >= 5, establishing a precise arithmetic framework for generating explicit solutions, which are further illustrated with numerical examples for p = 5, 7, 11, and 29.