Consecutive-Prime Products Among Binomial Coefficients

Let p₁=2 0,\[ p_a,p_b_ε(klog n)25/12+ε,\]and uniformly\[ log p_b(log n)2/3(loglog n)1/3.\]Consequently the support length $L=b-a+1$ satisfies the global lower bound\[ L k(log n/loglog n)1/3\]for all sufficiently large solutions. We also obtain an explicit globalalternative controlling the lower support endpoint, sharper bounds for fixedk, a classification of all solutions supported on at most three primes,finiteness for fixed k and fixed block length, and the counting estimate\[ #\{(n,k):n≤ X\} ≤exp\!(O((log X)2/3(loglog X)1/3)).\]For the two initial branches of the case $k=2$, namely pb#=n(n-1) and2\,pb#=n(n-1), we show that the admissible endpoints pb have densityzero among the primes, using an averaged Chebotarev density theorem ofLemke Oliver and Smith, and that under the generalised Riemann hypothesisthere are O(X1/2(log X)⁴) of them up to X. We verify that thesebranches have no further solutions with pb≤10¹¹, show that the twoequations are solvable for the same pb only when pb∈\3,7\, andsolve the analogous problem over Fq[T] completely.A pigeonhole argument shows that, at the uniform smoothness scaledisplayed above, smooth coprime pairs attain the extremal spacing, so thatspacing estimates for generic smooth coprime pairs cannot decide the problemuniformly in k; instead we isolate a linear-form hypothesis for $k=2$ anda smooth-tuple hypothesis for k≥3 that together imply finiteness.