PreprintCC BY-ND 4.0HAL
ON QUASI CLOSED CHAINS AND QUASI COMPLETE NUMBERS
Abstract
An addition chain of length h that leads to an integer n≥ 2 is a strictly increasing sequence of positive integers s0=1,s1=2,…,sh=n such that si=sj+sk with i>j≥ k≥ 0 for each i∈ {1,…,h}. We denote the minimal length of an addition chain that leads to an integer target · by ℓ(·). In this paper, we introduce the concept of \emph{quasi closed} addition chains and show that numbers n≥ 2 that admit a minimal-length quasi closed chains that are also a minimal-length addition (quasi complete numbers) satisfy the Scholz conjecture, precisely the inequality ℓ(2n−1)≤ n−1+ℓ(n).
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BibTeX
@misc{agama2026quasi,
title = {{ON QUASI CLOSED CHAINS AND QUASI COMPLETE NUMBERS}},
author = {Theophilus Agama},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05755806v1},
note = {CC BY-ND 4.0},
}
RIS
TY - UNPB
TI - ON QUASI CLOSED CHAINS AND QUASI COMPLETE NUMBERS
AU - Theophilus Agama
PY - 2026
DA - 2026/09/19
PB - HAL
UR - https://hal.science/hal-05755806v1
AB - An addition chain of length h that leads to an integer n≥ 2 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 with i>j≥ k≥ 0 for each i∈ {1,…,h}. We denote the minimal length of an addition chain that leads to an integer target · by ℓ(·). In this paper, we introduce the concept of \emph{quasi closed} addition chains and show that numbers n≥ 2 that admit a minimal-length quasi closed chains that are also a minimal-length addition (quasi complete numbers) satisfy the Scholz conjecture, precisely the inequality ℓ(2^n−1)≤ n−1+ℓ(n).
ER -