A First Digit Theorem For Powers of Perfect Powers - Hürlimann, Werner (Communications in Mathematics and Applications, 2014)
For any fixed power exponent, it is shown that the first digits of powers from perfect power numbers follow a generalized Benford law (GBL) with size-dependent exponent that converges asymptotically to a GBL with half of the inverse power exponent. In particular, asymptotically as the power goes to infinity these first digit sequences obey Benford's law. Moreover, we show the existence of a one-parametric size-dependent exponent function that converge to these GBL's and determine an optimal value that minimizes its deviation to two minimum estimators of the size-dependent exponent over the finite range of powers from perfect power numbers less than $10^{5m · s}$, $m = 2,…,6$, where $s = 1,2,3,4,5$ is a fixed power exponent.