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For $k\\geq 2$, put \\[\n  D(k)=\\gcd_{2\\leq q\\leq k+1}\\binom{qk}{k},\n  \\qquad n=k+1. \\] If $P$ is the largest prime-power component $p^a$ exactly dividing $n$, then the criterion asserts \\[\n  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. 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