Universal Invariants for \(\gcd(n^k - 1, n! - 1) > 1\)

For integers \(k\geq2\) and \(\lambda_1,\lambda_2\in\{-1,+1\}\), let \(S_k^{\lambda_1,\lambda_2}:=\{n \in \mathbb{N}\setminus \{0,1\} :\gcd(n^k+\lambda_1,n!+\lambda_2)>1\}\). This family arose from a MathOverflow problem concerning the existence of integers \(n\) satisfying \(\gcd(n^2+1,n!+1)>1\) and from the subsequent question of whether we can obtain arbitrarily many, but finitely many, solutions by varying the exponent and the signs. We concentrate on the case \(\lambda_1=\lambda_2=-1\) and introduce the notion of a universal invariant for an odd prime \(p\), namely an integer \(n\geq2\) such that \(\gcd(n^{pm}-1,n!-1)>1\) for every integer \(m\geq1\). We prove that this condition is equivalent to \(\gcd(n^p-1,n!-1)>1\), describe the possible prime divisors through roots of unity over finite fields, and recall Jonathan Love's explicit construction of a universal invariant for every odd prime \(p\geq5\), with the cubic case treated separately. As an application, we prove that \(|S_k^{-1,-1}|\geq \omega_{\mathrm{odd}}(k)\) for every integer \(k\geq2\), where \(\omega_{\mathrm{odd}}(k)\) denotes the number of distinct odd prime divisors of \(k\). Hence, the cardinalities of these sets are unbounded. This lower bound is deliberately coarse and constructive, as it is obtained by considering only universal invariants and is not intended to describe the exact cardinalities of these sets. We also discuss minimal universal invariants, their persistent common divisors, the special structure of the cubic case, and a heuristic of Will Sawin suggesting that there may be infinitely many cubic universal invariants, although they appear to be exceptionally sparse. Whether \(S_k^{-1,-1}\) is finite or infinite for any fixed exponent \(k\geq3\) remains an open problem.