CONSECUTIVE-PRIME PRODUCTS AMONG BINOMIAL COEFFICIENTS

Let $p_1=2 0$,\[ p_a,p_b\ll_\varepsilon(k\log n)^{25/12+\varepsilon},\]and uniformly\[ \log p_b\ll(\log n)^{2/3}(\log\log n)^{1/3}.\]Consequently the support length $L=b-a+1$ satisfies the global lower bound\[ L\gg k\left(\frac{\log n}{\log\log n}\right)^{1/3}\]for all sufficiently large solutions. We also obtain an explicit globalalternative controlling the lower support endpoint, sharper bounds for fixed$k$, a classification of all solutions supported on at most three primes,finiteness for fixed $k$ and fixed block length, and the counting estimate\[ \#\{(n,k):n\leq X\} \leq\exp\!\bigl(O((\log X)^{2/3}(\log\log X)^{1/3})\bigr).\]For the two initial branches of the case $k=2$, namely $p_b\#=n(n-1)$ and$2\,p_b\#=n(n-1)$, we show that the admissible endpoints $p_b$ have densityzero among the primes, using an averaged Chebotarev density theorem ofLemke Oliver and Smith, and that under the generalised Riemann hypothesisthere are $O(X^{1/2}(\log X)^4)$ of them up to $X$. We verify that thesebranches have no further solutions with $p_b\leq10^{11}$, show that the twoequations are solvable for the same $p_b$ only when $p_b\in\{3,7\}$, andsolve the analogous problem over $\mathbb F_q[T]$ 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 $k\geq3$ that together imply finiteness.