| viXra.org > Number Theory > viXra:2608.0029 |
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| viXra.org > Number Theory > viXra:2608.0029 |
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Number Theory |
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Authors: Theophilus Agama
An addition chain of length $h$ that leads to an integer $ngeq 2$ is a strictly increasing sequence of positive integers $s_0=1,s_1=2,ldots,s_h=n$ such that $s_i=s_j+s_k$ with $i>jgeq kgeq 0$ for each $iin {1,ldots,h}$. We denote the minimal length of an addition chain that leads to an integer target $cdot$ by $ell(cdot)$. In this paper, we introduce the concept of emph{quasi closed} addition chains and show that numbers $ngeq 2$ that admit a minimal-length quasi closed chains that are also a minimal-length addition (quasi complete numbers) satisfy the Scholz conjecture, precisely the inequality $$ ell(2^n-1)leq n-1+ell(n).$$
Comments: 10 Pages.
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