arXiv:2611.14600v1 Announce Type: new Abstract: 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 arbitrarily many, but finitely many, solutions can be obtained 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 $\gcd(n^{p\cdot m}-1,n!-1)>1$ for every integer $m\geq1$. We prove that this condition is equivalent to $\gcd(n^p-1,n!-1)>1$, characterize the possible prime divisors in terms of roots of unity over finite fields, and recall an explicit construction due to Jonathan Love that yields 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 as $k$ varies. This lower bound is deliberately coarse and constructive, as it is obtained by considering only universal invariants; stronger general lower bounds are obtained by different constructions. We also discuss minimal universal invariants, their persistent common divisors, the special structure of the cubic case, and a heuristic due to Will Sawin suggesting that there may be infinitely many cubic universal invariants, although the known examples are exceptionally sparse. Whether $S_k^{-1,-1}$ is finite or infinite for any fixed exponent $k\geq3$ remains an open problem.
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