# [Resource Topic] 2023/1959: On the notion of carries of numbers $2^n-1$ and Scholz conjecture

Welcome to the resource topic for 2023/1959

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

Authors: Theophilus Agama

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.

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