6 problems

Ufnarovski–Åhlander conjecture. For every $x\in\mathbb{N}$, exactly one of the following could happen: either $D^i(x)=0$ or $p^p$ for some prime $p$ for sufficiently large $i$, or

Ufnarovski–Åhlander conjecture. There exists an infinite sequence $(a_n)_{n\geq 1}$ of different natural numbers such that $a_1=1$ and

Ufnarovski–Åhlander conjecture. For every integer $n\geq 0$, there are infinitely many $x>0$ such that $x$ has exactly $n$ anti-derivatives.

Fix a prime $p$ and let $D_p$ denote the arithmetic partial derivative with respect to $p$, extended to rational numbers. For a nonzero rational number $x$, a rational number $y$ i…

Let $D:\mathbb{Z}\to\mathbb{Z}$ be the arithmetic derivative, and call an integer $y$ an integral anti-derivative of $x$ when $D(y)=x$. Let $i(x)$ denote the number of anti-derivat…

Small Derivatives Conjecture. There is an absolute constant $0<\eta<1$ such that for all but finitely many triples of coprime positive integers $(a,b,c)$ satisfying $a+b=c$ and not…