A Short Note on Gaps between Powers of Consecutive Primes

Let $\alpha, \beta \geq 0$ and $\alpha + \beta < 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^\beta(p_{n+1}^\alpha - p_n^\alpha) = 0$, where $p_n$ is the $n$th prime. This notes an improvement over results of S\'{a}ndor and gives additional evidence towards a conjecture of Andrica. This follows directly from recent results on prime pairs from Maynard, Tao, Zhang.