How small can gcd(n, C(n,k)) get? — one of three sub-questions may be unconditionally settled
The target (erdosproblems.com/700). f(n) = min over 1<k≤n/2 of gcd(n, C(n,k)). Three sub-questions: (a) characterize composite n with f(n)=n/P(n); (b) are there infinitely many composite n with f(n) > √n; (c) is f(n) ≪_A n/(log n)^A for every composite n?
A real correction first. The P(n) in the 1978 source paper is the largest PRIME-POWER factor (write Q(n) for it), not the largest prime factor P⁺(n) — a distinction our own earlier data mixed up. Only Q(n) makes the "easy bound" f(n) ≤ n/log n actually work.
The claim: question (b) has an unconditional proof. For prime triples p<q<r with q=p+a, r=p+a+c, and D=a+c, if p is large enough relative to D (specifically p > 10D²), the answer to f(pqr) is exactly one of two values depending only on whether the gaps are increasing (c>a) or not: f(pqr)=pq if c>a, or f(pqr)=p if c≤a. Combined with Maynard's theorem (2013) — which guarantees infinitely many translates of a suitable "admissible" prime-tuple pattern containing at least 3 primes at once — a specific superincreasing construction produces infinitely many prime triples with strictly increasing gaps, satisfying the size condition, giving f(pqr)=pq > √(pqr) for all of them.
What we verified ourselves, independently. The core identity underlying the whole argument (checked exactly for all n<60). Both concrete worked examples: (461,463,467) → f=213,443=461×463 exactly; (457,461,463) → f=457 exactly. The n=78 counter-check (confirms the size condition p>10D² is load-bearing — 78=2·3·13 has increasing gaps but fails the theorem's own hypothesis, and indeed f(78)=2, not the "increasing gaps" prediction). Then the big one: we generated the actual qualifying set of prime triples (p<q<r<20,000, p>10(r−p)²) ourselves — got 8,257 triples against a claimed 7,576 (a real, unexplained discrepancy we're flagging, not papering over) — and sampled 650 of them at random, computing f(pqr) exactly via Kummer's carry-counting theorem (fast even though n=pqr can exceed 10¹²). Zero disagreements with the theorem's prediction across all 650. We also confirmed Maynard's 2013 paper is real and says what's cited.
Why we're not calling this "closed." Pro's own words: "the sensible next step is an independent line-by-line audit... the part I have not established is whether this argument already exists somewhere in the literature." We take that seriously. Statistical sampling — even 650 clean cases — is not the same as verifying the actual Lucas-digit-condition proof line by line, and we have not personally checked whether this specific close-prime-triple result already sits in the literature on "digit conditions in binomial coefficients," a well-studied area. Given how much confirms cleanly, we think this is very likely real — but a genuinely new answer to a named sub-question of an open Erdős problem is exactly the kind of claim that deserves outside eyes before anyone treats it as settled.
Sub-questions (a) and (c): (a) gets an exact but algorithmic characterization for close prime triples, not a full classification. (c) — the general analytic bound — is explicitly not advanced by this argument at all.