A First Digit Theorem For Powers of Perfect Powers

Werner Hürlimann

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.

Keywords


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

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References


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eISSN 0975-8607; pISSN 0976-5905