Let$$f(n)=\underset{1<k\le n/2}{\min }\text{gcd}\left(n,\left(\genfrac{}{}{0ex}{}{n}{k}\right)\right).$$ Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)$ is the largest prime dividing $n$. Are there infinitely many composite $n$ such that $f(n)>n^{1/2}$? Is it true that, for every composite $n$,$$f(n){\ll }_A\frac{n}{(\log n{)}^A}$$for every $A>0$?

#700 RE-ASSESSED after reading erdosproblems comments (Tao, StijnC): substantially LESS attractive. Part 1 declared 'hopeless/undoable' by StijnC (confirms our probe). Part 3 'likely very hard' (Tao). Part 2 ALREADY has TWO conditional infinite families (StijnC + Kadi Siigur) -- the only open piece is an UNCONDITIONAL family, which StijnC found nontrivial AND says 'would not add much insight'. Recommend SKIP.

Computed f(n) via Kummer (gcd=prod_{p|n} p^min(v_p(n),v_p(C(n,k))), min over k). PART 1: characterize composite n with f(n)=n/P(n). 1026/1696 composite n<=2000 satisfy it. Structure (verified): f(n)=n/P(n) iff the best k can exclude ONLY the largest prime (carry at every smaller prime, no-carry at P), AND no k can simultaneously avoid carries at two primes. Holds for BALANCED primes; FAILS for spread primes -- 378 exceptions n<=1200 (e.g. n=78=2*3*13 f=2; n=110 f=2; n=170=2*5*17 f=5), all with a dominant large prime where some k kills >=2 factors. So part 1 is an intricate prime-SIZE-dependent characterization (Kummer/CRT), research-level. PART 2 (infinitely many composite n with f(n)>sqrt n): the 89 examples n<=2000 are all multi-prime with CLUSTERED factors and f(n)=n/P(n) (e.g. 30=2*3*5 f=6; 105=3*5*7 f=15; 385=5*7*11 f=35; 770=2*5*7*11 f=70). For >=3 clustered primes near x, n~x^3, P~x, f=n/P~x^2=n^(2/3)>sqrt(n). CONSTRUCTION-SHAPED & likely provable: exhibit an infinite family of balanced-prime n where f(n)=n/P(n) (Kummer argument: no k avoids carries at >=2 clustered primes), giving f>sqrt(n). NOTE: NOT every 3-prime n works (76 squarefree-3-prime exceptions <=3000, all SPREAD e.g. 78,110,170), so the family must be clustered/balanced. This is the genuinely Vela-shaped sub-question -- existence-by-construction, the shape that works (cf #488), NOT a universal/finiteness wall (cf #699/#931). PART 3 (f(n) << n/(log n)^A for all A): universal upper bound, analytic NT wall -- skip. SAME binomial-gcd/Kummer family as #699; Part-2 crux (f=n/P for clustered family) is a real carry/CRT argument but tractable. || COMMENTS (erdosproblems.com/700): Terence Tao (Oct 2025) -- literature review found nothing beyond Erdos-Szekeres; Guy B33 'Largest divisor of a binomial coefficient' related but does not reference this; ChatGPT's Guy-B33-specific claim was a hallucination. StijnC (Aug 2025, the same StijnC as #699) mapped it: PART 1 'likely undoable/seems hopeless' (3-prime f=n/P is erratic 'sometimes yes sometimes no'; depends intricately on prime sizes -- matches our probe's 378 exceptions). PART 3 'very hard' even for n^{1-eps}. PART 2 'should definitely be yes' with TWO explicit CONDITIONAL infinite families: (F1) q=2k+3 prime, k=4 mod 20 (OEIS A057732), n=q(q+1) => f(n)=q+1; (F2, Kadi Siigur) p,q primes 2^l<p<q<2^{l+1}, pq=1 mod 2^l, n=2pq => f(n)=2p. BOTH conditional on unproven prime-infinitude (special form / twin-like). The OPEN challenge = an UNCONDITIONAL family ('a different construction may do the work'), which is hard AND, per StijnC, 'would not add much to the insight the conditional proofs already gave'. NET: our independent probe reproduced StijnC's structural conclusions, but StijnC is ahead of us (explicit families); the residual open piece is hard + low-value. SKIP as a serious target.