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

TL;DR

This paper proves the Scholz conjecture for a new class of integers of the form n=2^m(23)+7, expanding the set of integers for which the conjecture is known to hold, based on previous work by Thurber.

Contribution

It extends the validity of the Scholz conjecture to specific integers of the form 2^m(23)+7 and related forms, providing new proofs for these cases.

Findings

The conjecture holds for n=2^m(23)+7.

The conjecture holds for integers of a form given by Thurber.

The paper confirms the conjecture for a new infinite set of integers.

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.