A Superlogarithmic Lower Bound for Prime-Free Intervals Following Even Powers of a Fixed Prime

We investigate the distance to the next prime following even powers of a fixed prime \(q\), defining \(K_q(n)=\operatorname{nextprime}(q^{2n})-q^{2n}\). Using Fermat's little theorem, a constructive least-common-multiple argument, and a known upper-bound sieve estimate, we establish that for every fixed prime \(q\), there exist infinitely many positive integers \(n\) satisfying \(K_q(n)\gg_q \dfrac{\log n\log\log n}{\log\log\log n}\). Consequently, \(\limsup_{n\to\infty}\dfrac{K_q(n)}{\log n}=\infty\). The result provides a quantitative construction of arbitrarily long sequences of consecutive composite integers immediately following numbers of the form \(q^{2n}\). It also establishes that no finite upper bound depending only on \(q\), including one of the form \(Cq^\alpha\) with absolute constants \(C>0\) and \(\alpha\in\mathbb R\), can universally bound the distance to the next prime for all positive integers \(n\). The note presents a self-contained derivation using established number-theoretic techniques and illustrates the construction with an explicit numerical example.