31 problems

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 $a,b,c$ 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

Iraids et al.'s conjecture.

Iraids et al.'s conjecture.

Let $S_u$ for $u\in\{0,1,2\}$ be the sets … ordered by $x\preccurlyeq y$ if and only if $x>y$. For a well-ordered set $L$ and an ordinal $\alpha$, let $L[\alpha]$ denote the elemen…

Let the sequences referred to in Conjecture 3 be sequences of natural numbers intended to describe numbers of complexity $3n+u$. Conjecture 3. There should be sequences satisfying…

Let $\|n\|$ denote the complexity of a natural number, and let $p,q,j$ be natural numbers. Corrected complexity conjecture. Under the additional hypotheses that $pq$ is stable and…

Let $\|n\|$ denote the complexity of a natural number. A number $n$ is stable if … for all $k\geq 0$. Stability conjecture. For every natural number $n$, there exists $K$ such that…

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 $(a_\alpha)$, $(b_\alpha)$, and $(c_\alpha)$ be the transfinite sequences of rational numbers, and let $\beta<\xi$. Suppose that $b_\beta=b/3^a$ with $\|b\|=3a+1$, and write…

Let $(a_\alpha)$, $(b_\alpha)$, and $(c_\alpha)$ be the transfinite sequences of rational numbers, and let $\beta<\xi$. Suppose that $a_\beta=b/3^a$ with $\|b\|=3a$, and write…

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 $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, 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

Let $a$ and $b$ be natural numbers. Arias's conjecture. There exists $K$ such that for all $k\ge K$, … This is one of the integer-complexity conjectures posed by the second author;…

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

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