A First Digit Theorem For Powers of Perfect Powers
DOI:
https://doi.org/10.26713/cma.v5i3.253Keywords:
first digit, perfect power number, asymptotic counting function, probabilistic number theory, mean absolute deviation, probability weighted least squaresAbstract
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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