  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
$\iota(2^n-1)$ producing $2^n-1$. Most notably, we show that if $2^n-1$ has
carries of degree at most $$\kappa(2^n-1)=\frac{1}{2}(\iota(n)-\lfloor
\frac{\log n}{\log 2}\rfloor+\sum \limits_{j=1}^{\lfloor \frac{\log n}{\log
2}\rfloor}\{\frac{n}{2^j}\})$$ then the inequality $$\iota(2^n-1)\leq n+1+\sum
\limits_{j=1}^{\lfloor \frac{\log n}{\log
2}\rfloor}\bigg(\{\frac{n}{2^j}\}-\xi(n,j)\bigg)+\iota(n)$$ holds for all $n\in
\mathbb{N}$ with $n\geq 4$, where $\iota(\cdot)$ denotes the length of the
shortest addition chain producing $\cdot$, $\{\cdot\}$ 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.
