Digits of powers of 2 in ternary numeral system

Yagub N. Aliyev
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 3, Pages 474–485
DOI: 10.7546/nntdm.2023.29.3.474-485
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Authors and affiliations

Yagub N. Aliyev
School of IT and Engineering, ADA University
Ahmadbey Aghaoglu str. 61, Baku, AZ1008, Azerbaijan


We study the digits of the powers of 2 in the ternary number system. We propose an algorithm for doubling numbers in ternary numeral system. Using this algorithm, we explain the appearance of “stairs” formed by 0s and 2s when the numbers 2^n (n=0,1,2, \ldots) are written vertically so that for example the last digits are forming one column, the second last digits are forming another column, and so forth. We use the patterns formed by the leftmost digits, and the patterns formed by the rightmost digits to prove that the sizes of these blocks of 0s and 2s are unbounded. We also study how this regularity changes when the digits are taken between the left end and the right end of the numbers.


  • Ternary numeral system
  • Benford’s law
  • Digits
  • Zeros
  • Twos
  • Powers of two

2020 Mathematics Subject Classification

  • Primary: 11A63
  • Secondary: 11A07, 11B83


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Manuscript history

  • Received: 23 November 2022
  • Revised: 22 June 2023
  • Accepted: 27 June 2023
  • Online First: 10 July 2023
  • Correction Notice: 24 July 2023
    Per Author’s request made on 23 July 2023, the following correction is made on page 474, line -7 of the Online First (electronic) version:
    Old: In 1979, Paul Erdős conjectured that there are only finitely many positive integers that can be written as sum of distinct powers of 2.
    New: In 1979, Paul Erdős conjectured that there are only finitely many positive integers n such that 2^n can be written as sum of distinct powers of 3.

Copyright information

Ⓒ 2023 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Aliyev, Y. N. (2023). Digits of powers of 2 in ternary numeral system. Notes on Number Theory and Discrete Mathematics, 29(3), 474-485, DOI: 10.7546/nntdm.2023.29.3.474-485.

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