21 problems

Let $a,b,c$ be nonnegative integers satisfying $a+b+c>0$ and $c<6$. For a positive integer $n$, let $\|n\|$ denote its integer complexity, namely the least number of ones needed to…

2-complexity conjecture. For $m\geq0$ and $r\geq1$,

Integer-complexity conjecture. For $a\geq 1$ and $b\geq 0$,

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…

Collapse conjecture. If

The efficient-prime finite-control conjecture. For any $\alpha>\frac{2}{\log 2}$, there is a finite set of primes $S_\alpha$ such that, for any $m$, if $v_p$ is sufficiently large…

The balanced and efficient numbers counting conjecture. For all $x$,

The 1439 extremal conjecture. For all $n>1$, one has

Let $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, and $(c_\alpha)_{\alpha<\xi}$ be the transfinite sequences associated with the three complexity classes modulo $3$. Block-…

Let $\|n\|$ denote the complexity of a natural number, and define … Let $(a_\alpha)_{\alpha<\xi}$ be the transfinite sequence associated with numbers of complexity divisible by $3$…

Let $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, and $(c_\alpha)_{\alpha<\xi}$ be the transfinite sequences of rational numbers from the preceding conjecture. Denominator…

Let $\|n\|$ denote the complexity of a natural number, and let $\xi$ be an ordinal. Consider three transfinite sequences $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, and…

Let $\|n\|$ denote the complexity of a natural number, and let $p,q$ be natural numbers. Product complexity conjecture. For each pair of natural numbers $p$ and $q$, there exists…

Arias's conjecture. The set $A$ has finite symmetric difference with $E$, and the one-sided difference $A\setminus E$ is a finite subset of

Sequence-growth conjecture. For all $k\in\mathbb N$,

Powers-of-three complexity conjecture. For every $k\in\mathbb N$,

Asymptotic complexity conjecture.

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 $\|n\|$ denote the smallest number of ones needed to write the positive integer $n$ using addition and multiplication. For integers $k,\ell\geq 0$, with $k$ and $\ell$ not both…

For each positive integer $n$, let $\Vert n\Vert$ be the least number of $1$'s needed to represent $n$ using addition and multiplication. The quantity $\Vert n-1\Vert-\Vert n\Vert$…

Let $\|n\|$ denote the least number of ones needed to write the positive integer $n$ using addition, multiplication, and parentheses. For integers $a,k\geq 0$ with $a+k\geq 1$, con…