@article{Tarek2026, 
author = {S. Tarek and Hatem M. Bahig and M. Anwar},
title = {Addition-multiplication chains with one small step},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {16235-16263},
keywords = {addition chains, addition-multiplication chains, shortest length, small step},
url = {https://www.sciopen.com/article/10.3934/Math.2026667},
doi = {10.3934/Math.2026667},
abstract = {An addition-multiplication chain of length    l for a positive integer    n is a monotonic increasing sequence    1  =      a    0    <      a    1    <  …  <      a    l    =  n of positive integers, such that for each    1  ≤  i  ≤  l  , we have        a    i    =            a      j        +            a      k          or        a    i    =            a      j        ×            a      k        , where    0  ≤  k  ≤  j  ≤  i  −  1. In this paper, we establish upper bounds for each element in any AM-chain based on the number of squaring steps and the distribution of non-squaring steps preceding the given element. Moreover, we get additional upper bounds independent of the distribution of non-squaring steps. These bounds yield a new upper bound for the number of non-squaring steps in any AM-chain. Finally, we determine all AM-chains that include exactly one small step and, as a consequence, compute the shortest lengths        ℓ          A      M        (  .  ) of certain integers.}
}