## Abstract

In this article we give an informal presentation of the so-called Benford’s law, a counterintuitive probability distribution that describes the frequency with which the first significant digits—that is, a digit between 1 and 9—appear in sets of numbers associated with heterogeneous quantities. Unlike what might be expected, these frequencies are not equal but are decreasing, that is, 1 appears more often than 2, which appears more often than 3 and so on. After an introduction to the law, we will show how, perhaps unexpectedly, it is satisfied by the numbers generated by some sequences of powers. We will then illustrate some applications in the socio-economic field (for instance, to identify accounting or electoral fraud). Finally we will use it to analyse the tax returns of some former US presidential candidates.

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## Notes

- 1.
The partial volume effect is responsible for the reduced accuracy with which the boundaries of small formations can be determined in clinical images.

- 2.
This analysis could not be performed for the incumbent president of the USA, Donald J. Trump, since his tax return for the year immediately preceding his inauguration was not made available.

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## Additional information

The authors thank the Centro di Economia Quantitativa of the Ca’ Foscari University of Venice (Italy) for the support received.

## Appendix: Benford’s law

### Appendix: Benford’s law

Benford’s law, in its more complete form, claims that the probability that the *i*-th digit \(D_i\) equals \(d_i\) for \(i = 1, \ldots , n\), that is, for the first *n* base-10 digits, is

Moreover, Benford’s law has several interesting characterisations.

First, a sequence \((x_n)_n\) satisfies Benford’s law if and only if the sequence \((\{log_{10}\left| x_n\right| \})_n\) is equidistributed in the interval [0, 1), where \(\{x \}\) is the fractional part of *x*.

Then, when the numbers we are considering are given by physical measurements, Benford’s law does not depend on the units chosen (*scale invariance*). Moreover, it does not depend on the base chosen to represent the numbers either (*base invariance*).

For further information we refer the interested reader to Theodore P. Hill’s article [9].

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Corazza, M., Ellero, A. & Zorzi, A. The importance of being “one” (or Benford’s law).
*Lett Mat Int* **6, **33–39 (2018). https://doi.org/10.1007/s40329-018-0218-4

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### Keywords

- Benford’s law
- First significant digit
- Power sequences
- Tax evasion
- Vote-rigging