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A first digit theorem for powerful integer powers.

Werner Hürlimann1

  • 1Swiss Mathematical Society, University of Fribourg, 1700 Fribourg, Switzerland.

Springerplus
|November 7, 2015
PubMed
Summary
This summary is machine-generated.

The first digits of integer powers follow a generalized Benford law (GBL). As powers increase, these digits increasingly obey the standard Benford's law, with a specific size-dependent exponent function identified.

Keywords:
Asymptotic counting functionFirst digitMean absolute deviationPowerful numberProbabilistic number theoryProbability weighted least squares

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Area of Science:

  • Number Theory
  • Digital Analysis
  • Statistical Distributions

Background:

  • Benford's Law describes the frequency distribution of first digits in many real-life datasets.
  • Generalized Benford Laws (GBLs) extend this concept to more complex data sequences.
  • The behavior of first digits in sequences of integer powers is a topic of ongoing mathematical interest.

Purpose of the Study:

  • To investigate the distribution of first digits in sequences of powerful integer powers.
  • To establish the convergence of these distributions to a Generalized Benford Law.
  • To determine a size-dependent exponent function and its optimal parameters.

Main Methods:

  • Analysis of sequences of powerful integer powers for a fixed power exponent.
  • Asymptotic analysis to determine limiting behavior as powers approach infinity.
  • Development and optimization of a one-parametric size-dependent exponent function.

Main Results:

  • Demonstrated that first digits of powerful integer powers follow a GBL with a size-dependent exponent.
  • Established asymptotic convergence to a GBL with the inverse double power exponent.
  • Showed that for large powers, the sequences obey the standard Benford's Law.
  • Identified an optimal parameter for the size-dependent exponent function.

Conclusions:

  • The study provides a comprehensive analysis of first-digit distributions in integer powers.
  • Confirms the applicability of Generalized Benford Laws to these sequences.
  • Offers insights into the mathematical properties of number sequences and their statistical behavior.