On addition chains and progress on the Scholz conjecture
Theophilus Agama
Abstract
In this paper, we develop some new classes of methods to study the Scholz conjecture on addition chains. It turns out that the exponents of numbers of the form $2^n-1$ largely determine the length of the shortest addition chain for number producing $2^n-1$. Exploiting the notion of carries, we obtain improved upper bounds for the length of the shortest addition chains $ι(2^n-1)$ producing $2^n-1$. Most notably, we show that if $2^n-1$ has carries of degree at most $$κ(2^n-1)=\frac{1}{2}(ι(n)-\lfloor \frac{\log n}{\log 2}\rfloor+\sum \limits_{j=1}^{\lfloor \frac{\log n}{\log 2}\rfloor}\{\frac{n}{2^j}\})$$ then the inequality $$ι(2^n-1)\leq n+1+\sum \limits_{j=1}^{\lfloor \frac{\log n}{\log 2}\rfloor}\bigg(\{\frac{n}{2^j}\}-ξ(n,j)\bigg)+ι(n)$$ holds for all $n\in \mathbb{N}$ with $n\geq 4$, where $ι(\cdot)$ denotes the length of the shortest addition chain producing $\cdot$, $\{\cdot\}$ denotes the fractional part of $\cdot$ and where $ξ(n,1):=\{\frac{n}{2}\}$ with $ξ(n,2)=\{\frac{1}{2}\lfloor \frac{n}{2}\rfloor\}$ and so on.