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

Abstract

The Scholz conjecture on addition chains states that $\ell(2^n-1) \leq \ell(n) + n -1$ for all integers $n$ where $\ell(n)$ 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 $\ell(2n) = \ell(n)$. 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.