Paper 2023/1959

On the notion of carries of numbers $2^n-1$ and Scholz conjecture

Abstract

Applying the pothole method on the factors of numbers of the form $2^n-1$, we prove that if $2^n-1$ has carries of degree at most $$\kappa(2^n-1)=\frac{1}{2(1+c)}\lfloor \frac{\log n}{\log 2}\rfloor-1$$ for $c>0$ fixed, then the inequality $$\iota(2^n-1)\leq n-1+(1+\frac{1}{1+c})\lfloor\frac{\log n}{\log 2}\rfloor$$ holds for all $n\in \mathbb{N}$ with $n\geq 4$, where $\iota(\cdot)$ denotes the length of the shortest addition chain producing $\cdot$. In general, we show that all numbers of the form $2^n-1$ with carries of degree $$\kappa(2^n-1):=(\frac{1}{1+f(n)})\lfloor \frac{\log n}{\log 2}\rfloor-1$$ with $f(n)=o(\log n)$ and $f(n)\longrightarrow \infty$ as $n\longrightarrow \infty$ for $n\geq 4$ then the inequality $$\iota(2^n-1)\leq n-1+(1+\frac{2}{1+f(n)})\lfloor\frac{\log n}{\log 2}\rfloor$$ holds.

Note: This paper is quite technical and it uses the newly introduced notion of carries to obtain improved upper bounds for the length of the shortest addition chains producing numbers of the form $2^n-1$.