The Scholz conjecture for $n=2^m(23)+7$, $m\in {\mathbb{N}}^{\ast }$

The Scholz conjecture on addition chains states that $\mathrm{\ell }(2^n-1)\le \mathrm{\ell }(n)+n-1$ for all integers $n$ where $\mathrm{\ell }(n)$ stands for the minimal length of all addition chains for $n$. It is proven to hold for infinite sets of integers. In this paper, we will prove that the conjecture still holds for $n=2^m(23)+7$. It is the first set of integers given by Thurber \cite{9} to prove that there are an infinity of integers satisfying $\mathrm{\ell }(2n)=\mathrm{\ell }(n)$. Later on, Thurber \cite{4} give a second set of integers with the same properties ($n=2^{2m+k+7}+2^{2m+k+5}+2^{m+k+4}+2^{m+k+3}+2^{m+2}+2^{m+1}+1$). We will prove that the conjecture holds for them as well.