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PreprintCC BY-ND 4.0HAL

ON A PROBABILISTIC ITERATED FACTOR METHOD

  • Theophilus Agama
Open at sourcePDFCite

Abstract

We introduce a probabilistic version of the iterated factor method introduced in our previous investigations. Let n be drawn uniformly from {1,2,…,M}. Set s:=s(n)= ⌊ √(log2n)/(log log2n) ⌋ and t(n)=√(log log n) and kn= ⌊ n/2 ⌋s. For N⟶ ∞ with Zn∼ N(0,1)~(normally distributed) and additionally that Pr[ν(kn)≤ s and |Zn|≤ t]⟶ c for 0≤ c≤ 1, we show that the inequality ℓ(2n−1)≤ n−1+log2n+C√(log2n/(log log2n)) for some absolute constant C>0 with probability that tends to c for 0≤ c≤ 1. This breaks the O((log n)/(log log n)) barrier that is guaranteed to be achieved using the Brauer method in this randomized setting. This result can be seen as injecting probabilistic methods into the theory of addition chains.

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Cite

BibTeXRIS
BibTeX
@misc{agama2026probabilistic,
  title = {{ON A PROBABILISTIC ITERATED FACTOR METHOD}},
  author = {Theophilus Agama},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05784795v1},
  note = {CC BY-ND 4.0},
}
RIS
TY  - UNPB
TI  - ON A PROBABILISTIC ITERATED FACTOR METHOD
AU  - Theophilus Agama
PY  - 2026
DA  - 2026/10/07
PB  - HAL
UR  - https://hal.science/hal-05784795v1
AB  - We introduce a probabilistic version of the iterated factor method introduced in our previous investigations. Let n be drawn uniformly from {1,2,…,M}. Set s:=s(n)= ⌊ √(log_2n)/(log log_2n) ⌋ and t(n)=√(log log n) and k_n= ⌊ n/2 ⌋_s. For N⟶ ∞ with Z_n∼ N(0,1)~(normally distributed) and additionally that Pr[ν(k_n)≤ s and |Z_n|≤ t]⟶ c for 0≤ c≤ 1, we show that the inequality ℓ(2^n−1)≤ n−1+log_2n+C√(log_2n/(log log_2n)) for some absolute constant C>0 with probability that tends to c for 0≤ c≤ 1. This breaks the O((log n)/(log log n)) barrier that is guaranteed to be achieved using the Brauer method in this randomized setting. This result can be seen as injecting probabilistic methods into the theory of addition chains.
ER  -