# A First Digit Theorem For Powers of Perfect Powers

## DOI:

https://doi.org/10.26713/cma.v5i3.253## Keywords:

first digit, perfect power number, asymptotic counting function, probabilistic number theory, mean absolute deviation, probability weighted least squares## Abstract

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 $10^{5m \cdot s}$, $m = 2,\ldots,6$, where $s = 1,2,3,4,5$ is a fixed power exponent.### Downloads

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## How to Cite

*Communications in Mathematics and Applications*,

*5*(3), 91–99. https://doi.org/10.26713/cma.v5i3.253

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