A note on the Aiello–Subbarao conjecture on addition chains

Hatem M. Bahig
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 2, Pages 276–280
DOI: 10.7546/nntdm.2022.28.2.276-280
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Hatem M. Bahig
Department of Mathematics, Faculty of Science, Ain Shams University
Cairo, Egypt

Abstract

Given a positive integer x, an addition chain for x is an increasing sequence of positive integers 1=c_0,c_1, \ldots , c_n=x such that for each 1\leq k\leq n, c_k=c_i+c_j for some 0\leq i\leq j\leq k-1. In 1937, Scholz conjectured that for each positive integer x, \ell(2^x-1) \leq \ell(x)+ x-1, where \ell(x) denotes the minimal length of an addition chain for x. In 1993, Aiello and Subbarao stated the apparently stronger conjecture that there is an addition chain for 2^x-1 with length equals to \ell(x)+x-1 . We note that the Aiello–Subbarao conjecture is not stronger than the Scholz (also called the Scholz–Brauer) conjecture.

Keywords

  • Addition chain
  • Aiello–Subbarao’s conjecture
  • Scholz–Brauer’s conjecture

2020 Mathematics Subject Classification

  • 11Y16
  • 11Y55

References

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Manuscript history

  • Received: 21 January 2021
  • Revised: 14 April 2022
  • Accepted: 9 May 2022
  • Online First: 10 May 2022

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Cite this paper

Bahig, H. M. (2022). A note on the Aiello–Subbarao conjecture on addition chains. Notes on Number Theory and Discrete Mathematics, 28(2), 276-280, DOI: 10.7546/nntdm.2022.28.2.276-280.

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