A First Digit Theorem For Powers of Perfect Powers

Werner Hürlimann

doi:10.26713/cma.v5i3.253

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

Full Text:

PDF

References

N.H.F. Beebe (2014), A bibliography of publications about Benford’s law, Heap’s law, and Zipf’s law, version 1.60. URL: ftp://ftp.math.utah.edu/pub/tex/bib/benfords-law.pdf

F. Benford (1938), The law of anomalous numbers, Proc. Amer. Phil. Soc. 78, 551-572.

A. Berger and T. Hill (2009), Benford Online Bibliography, URL: http://www.benfordonline.net/

W. Hürlimann (2004), Integer powers and Benford’s law, Int. J. Pure Appl. Math. 11(1), 39-46.

W. Hürlimann (2009), Generalizing Benford’s law using power laws: applications to integer sequences, Int. J. Math. and Math. Sci., Article ID 970284.

W. Hürlimann (2014), A first digit theorem for square-free integer powers, Pure Mathematical Sciences 3(3), 129-139.

R. Jakimczuk (2011), On the distribution of perfect powers, J. Integer Sequences 14, Article 11.8.5.

R. Jakimczuk (2013), Exact formulae for the perfect power counting function and the n-th perfect power, Int. Math. Forum 8(11), 513-516.

L.M. Leemis, B.W. Schmeiser and D.L. Evans (2000), Survival distributions satisfying Benford’s law, The Amer. Statistician 54(3), (2000), 1-6.

B. Luque and L. Lacasa (2009), The first-digit frequencies of prime numbers and Riemann zeta zeros, Proc. Royal Soc. A 465, 2197-2216.

S. Newcomb (1881), Note on the frequency of use of the different digits in natural numbers, Amer. J. Math. 4, 39-40.

M.J. Nigrini (2000), Digital analysis using Benford’s law: test statistics for auditors, Vancouver, Canada, Global Audit Publications.

M.J. Nigrini (2012), Benford’s Law. Applications for forensic accounting, auditing, and fraud detection, J. Wiley & Sons, Hoboken, New Jersey.

M.A. Nyblom (2006), A counting function for the sequence of perfect powers, Austr. Math. Soc. Gaz. 33, 338-343.

M.A. Nyblom (2008), Counting the perfect powers, Math. Spectrum 41, 27-31.

DOI: http://dx.doi.org/10.26713%2Fcma.v5i3.253

Refbacks

eISSN 0975-8607; pISSN 0976-5905