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PreprintCC BY-ND 4.0HAL

AN EXTENDED STRUCTURAL SCHOLZ RESULT ON ADDITION CHAINS

  • Theophilus Agama
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Abstract

An addition chain of length h that leads to a number n is a strictly increasing sequence of positive integers s0=1,s1=2,…,sh=n such that si=sj+sk~(i>j≥ k) for each 1≤ i≤ h. The \emph{Brauer} addition chain is the one where j=i−1 for each 1≤ i≤ h. Let ℓ(·) denote the minimal length of an addition chain that leads to an integer ·. We say that step i is \emph{Brauer} when j=i−1. Similarly, we say that step i is \emph{non-Brauer} when j<i−1. Under the condition that an integer n admits an optimal addition chain satisfying the conditions: there exists a consecutive \emph{non-Brauer} step (index) i−1<i with corresponding terms in the chain whose dominant summands have nondecreasing indices with the one with the greatest magnitude of Brauer index, and that a less dominant term of summands that calculate a term in the chain that never appears as a dominant term when calculating other terms in the chain has a corresponding \emph{Brauer} index or a succeeding \emph{Brauer} index, we have the inequality ℓ(2n−1)≤ n−1+ℓ(n).

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Cite

BibTeXRIS
BibTeX
@misc{agama2026extended,
  title = {{AN EXTENDED STRUCTURAL SCHOLZ RESULT ON ADDITION CHAINS}},
  author = {Theophilus Agama},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05756234v1},
  note = {CC BY-ND 4.0},
}
RIS
TY  - UNPB
TI  - AN EXTENDED STRUCTURAL SCHOLZ RESULT ON ADDITION CHAINS
AU  - Theophilus Agama
PY  - 2026
DA  - 2026/09/20
PB  - HAL
UR  - https://hal.science/hal-05756234v1
AB  - An addition chain of length h that leads to a number n is a strictly increasing sequence of positive integers s_0=1,s_1=2,…,s_h=n such that s_i=s_j+s_k~(i>j≥ k) for each 1≤ i≤ h. The \emph{Brauer} addition chain is the one where j=i−1 for each 1≤ i≤ h. Let ℓ(·) denote the minimal length of an addition chain that leads to an integer ·. We say that step i is \emph{Brauer} when j=i−1. Similarly, we say that step i is \emph{non-Brauer} when j<i−1. Under the condition that an integer n admits an optimal addition chain satisfying the conditions: there exists a consecutive \emph{non-Brauer} step (index) i−1<i with corresponding terms in the chain whose dominant summands have nondecreasing indices with the one with the greatest magnitude of Brauer index, and that a less dominant term of summands that calculate a term in the chain that never appears as a dominant term when calculating other terms in the chain has a corresponding \emph{Brauer} index or a succeeding \emph{Brauer} index, we have the inequality ℓ(2^n−1)≤ n−1+ℓ(n).
ER  -