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PreprintCC BY-ND 4.0HAL

THE SCHOLZ CONJECTURE IS TRUE FOR 2 n -1 FOR ALMOST ALL n

  • Theophilus Agama
Open at sourcePDFCite

Abstract

An addition chain of length h that leads to a number n is a sequence of positive integers s0=1,s1=2,…,sh=n such that si=sj+sk~(i>j≥ k) for each 1≤ i≤ h. A Brauer addition chain is the one where j=i−1 for each 1≤ i≤ h. Let ℓ(·) and ℓ*(·) denote the minimal length of an addition chain and the Brauer addition chain, respectively, that leads to an integer ·. Applying probabilistic methods to the \emph{iterated factor} method, we show that ℓ(2n−1)≤ n−1+ℓ(n) for almost all positive integers n as n⟶ ∞.

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Cite

BibTeXRIS
BibTeX
@misc{agama2026scholz,
  title = {{THE SCHOLZ CONJECTURE IS TRUE FOR 2 n -1 FOR ALMOST ALL n}},
  author = {Theophilus Agama},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05744549v1},
  note = {CC BY-ND 4.0},
}
RIS
TY  - UNPB
TI  - THE SCHOLZ CONJECTURE IS TRUE FOR 2 n -1 FOR ALMOST ALL n
AU  - Theophilus Agama
PY  - 2026
DA  - 2026/09/09
PB  - HAL
UR  - https://hal.science/hal-05744549v1
AB  - An addition chain of length h that leads to a number n is a 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. A Brauer addition chain is the one where j=i−1 for each 1≤ i≤ h. Let ℓ(·) and ℓ^*(·) denote the minimal length of an addition chain and the Brauer addition chain, respectively, that leads to an integer ·. Applying probabilistic methods to the \emph{iterated factor} method, we show that ℓ(2^n−1)≤ n−1+ℓ(n) for almost all positive integers n as n⟶ ∞.
ER  -