A Greatest Common Divisor Criterion of Certain Binomial Coefficients

Abstract: The binomial greatest common divisor (gcd) criterion recorded as OEIS A080170 is proven. The criterion also appears as conjecture (17) in Ralf Stephan's list of OEIS conjectures. For $k\geq 2$, put [ D(k)=\gcd_{2\leq q\leq k+1}\binom{qk}{k}, \qquad n=k+1. ] If $P$ is the largest prime-power component $p<sup>a$ exactly dividing $n$, then the criterion asserts [ D(k)=1 \quad\Longleftrightarrow\quad \frac{n}{P}>P. ] The proof is formalized in Lean and the Lean artifact is accepted as part of the Formal Conjectures project. Both the natural-language proof and the Lean formalization are generated by the MechMath Agent Team, an AI agent developed by the authors.