Simultaneous Prime-Free Runs After Even Powers with Prime Indices

For any fixed finite nonempty set \(\mathcal Q\) of primes, this note proves that infinitely many prime indices \(r\) simultaneously produce long runs of composite integers immediately after every \(q^{2r}\), with \(q\in\mathcal Q\). Define \(K_{\mathcal Q}(r)=\min_{q\in\mathcal Q}\bigl(\operatorname{nextprime}(q^{2r})-q^{2r}\bigr)\), where \(\operatorname{nextprime}(x)\) denotes the least prime strictly greater than \(x\). We prove that \(K_{\mathcal Q}(r)\gg_{\mathcal Q}\frac{\log r\,\log\log r}{\log\log\log r}\) along an unbounded set of prime indices. Consequently, for every prescribed positive integer \(B\), infinitely many primes \(r\) make all the integers \(q^{2r}+1,\ldots,q^{2r}+B\) composite simultaneously for every \(q\in\mathcal Q\). The same prime index works for all bases in the fixed finite set. The index \(r\) is prime, while the exponent \(2r\) is even. The proof combines an elementary congruence construction with Linnik's theorem on the least prime in an arithmetic progression and Kalmynin's sieve estimate for \(\log\operatorname{lcm}\{p-1:p\le x,\ p\text{ prime}\}\). The note includes a complete derivation from these established inputs and an explicit simultaneous example. The novelty of the exact formulation has not been established, and no resolution of a named open conjecture is claimed.