6 problems

Let $\|n\|_{2}$ denote the binary complexity of the positive integer $n$. Let $r$ be a positive integer and let $s$ be a nonnegative integer, with $(r,s)\ne(1,0)$. The binary analo…

Addition-chain defect order-type conjecture. For each whole number $k$, the set $\mathscr{D}^{\ell}\cap[0,k]$ has order type $\omega^k$. This conjecture concerns the fine structure…

Scholz's conjecture reduction. The following inequality holds:

Order-type conjecture for addition chain defects. For each whole number $k$, the set $\mathscr{D}^\ell\cap[0,k]$ has order type $\omega^k$.

Let $f^\ell(k)$ and $f^\ell_{\mathrm{st}}(k)$ be the lower-bound functions for ordinary and stable addition-chain defects, and let $f^*(k)$ and $f^*_{\mathrm{st}}(k)$ be the corres…

For a positive integer $n$, let $\nu_2(n)$ denote the number of $1$'s in its binary expansion, and let $s(n)$ denote the addition-subtraction chain length used in the paper. KnuthS…