On addition chains and progress on the Scholz conjecture

TL;DR

This paper introduces new methods to analyze addition chains and makes progress on the Scholz conjecture by deriving improved upper bounds for the length of shortest addition chains for numbers of the form 2^n-1.

Contribution

The paper develops carry analysis techniques to establish new bounds on addition chain lengths, advancing understanding of the Scholz conjecture.

Findings

Derived improved upper bounds for 5(2^n-1)

Showed the significance of the exponents of numbers 2^n-1 in addition chain length

Established conditions under which the bounds hold for all n  4

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 the number that leads to $2^n-1$. Using the carry analysis, we obtain improved upper bounds for the length of the shortest addition chains $\ell(2^n-1)$ producing $2^n-1$. In particular, we show that if $2^n-1$ has carry of degree at most $$ \kappa(2^n-1)=\frac{1}{2}\left(\ell(n)-\left\lfloor\frac{\log n}{\log 2}\right\rfloor+\sum \limits_{j=1}^{\lfloor \frac{\log n}{\log 2}\rfloor}\left\{\frac{n}{2^j}\right\}\right) $$ then $$ \ell(2^n-1)\leq n+1+\sum \limits_{j=1}^{\lfloor\frac{\log n}{\log 2}\rfloor}\bigg(\left\{\frac{n}{2^j}\right\}-\xi(n,j)\bigg)+\ell(n) $$ for all $n\in \mathbb{N}$ with $n\geq 4$, where $\ell(\cdot)$ denotes the length of the shortest addition chain that leads to $\cdot$, $\{\cdot\}$ denotes the fractional part of $\cdot$ and where $\xi(n,1):=\{\frac{n}{2}\}$ with $\xi(n,2)=\{\frac{1}{2}\lfloor \frac{n}{2}\rfloor\}$ and so on.