论文标题
关于加法链的分布
On the distribution of addition chains
论文作者
论文摘要
在本文中，我们研究了产生任何给定数量$ n \ geq 3 $的加成链的理论。为了估计加法链的部分总和，我们介绍了确定链和加法链的调节器的概念，并证明以下身份\ begin {align} \ sum \ limits_ {j = 2}^{δ（n）+1} s_j = 2（n-1）+（δ（n）-1）+a_ {Δ（n）} -r_ {Δ（n）+a {Δ（n）+1}+\ int+\ int \ int \ int \ int \ limits_} t} r_jdt。\ nonumber \ end {align}其中\ begin {align} 2，s_3 = a_3+r_3，\ ldots，s_ {δ（n）} = a_ {δ（n）}+r_ {δ（n）}，s_ {δ（Δ（n）+1} = a__ {Δ链条中的$ 1,2，\ ldots，s_ {δ（n）}，s_ {δ（n）+1} = n $的长度$δ（n）$。另外，我们获得了身份\ begin {align} \ sum \ limits_ {j = 2}^{δ（n）+1} a_j =（n-1）+（δ（n）-1）+a_ {δ（Δ（n）} - n）} - r_ {Δ \ limits_ {2 \ leq j \ leq t} r_jdt。\ nonumber \ end {align}
In this paper, we study the theory of addition chains producing any given number $n\geq 3$. With the goal of estimating the partial sums of addition chains, we introduce the notion of the determiners and the regulators of an addition chain and prove the following identities \begin{align}\sum \limits_{j=2}^{δ(n)+1}s_j=2(n-1)+(δ(n)-1)+a_{δ(n)}-r_{δ(n)+1}+\int \limits_{2}^{δ(n)-1}\sum \limits_{2\leq j\leq t}r_jdt.\nonumber \end{align}where \begin{align} 2,s_3=a_3+r_3,\ldots,s_{δ(n)}=a_{δ(n)}+r_{δ(n)},s_{δ(n)+1}=a_{δ(n)+1}+r_{δ(n)+1}=n\nonumber \end{align}are the associated generators of the chain $1,2,\ldots,s_{δ(n)},s_{δ(n)+1}=n$ of length $δ(n)$. Also we obtain the identity \begin{align} \sum \limits_{j=2}^{δ(n)+1}a_j=(n-1)+(δ(n)-1)+a_{δ(n)}-r_{δ(n)+1}+\int \limits_{2}^{δ(n)-1}\sum \limits_{2\leq j\leq t}r_jdt.\nonumber \end{align}
