24 problems

For a prime $p$, let $s_p(n)$ be the sum of the base-$p$ digits of $n$, let $\omega(n)$ count the distinct prime divisors of $n$, and define … Kellner's conjecture. The following s…

For $x\in\mathbb{N}$ and $n\geq 3$, define … where $s_{n-1}(r)$ is the digit sum of $r$ in base $n-1$. Moser's conjecture. For every $x\in\mathbb{N}$, … Here $S_{3,\;0}(x)$ is the…

A base-$g$ Niven number is a natural number divisible by the sum of its base-$g$ digits. A set of integers is an asymptotic basis of order $k$ if every sufficiently large natural n…

Digit-sum exception conjecture. No exceptions exist beyond the repdigit PINNs, the trivial PINNs $a0_{(0)}$ with $a\in\{1,2,3,4,5,6,7,8,9\}$, and the PINNs whose digit sums satisfy…

Fix a base $b\geq 2$. Write every $u\geq 1$ uniquely as a tower of powers of $b$ ending in $0<\omega\leq 1$, and let $\operatorname{ht}(u)$ be the height of this tower. Let $K(n)$…

For base $10$, let $s(v)$ be the sum of the decimal digits of $v$, and let $K(n)$ be the smallest number with exactly $n$ generators, meaning solutions $v$ of $u=v+s(v)$. Narasinga…

Let $K(n)$ denote the smallest base-$10$ number with exactly $n$ generators, where a generator of $u$ is a number $v$ satisfying $u=v+s(v)$ and $s(v)$ is the sum of the base-$10$ d…

Digit-sum strengthening. For every integer $k\geq 9$,

A power of two is an integer of the form $2^n$ with $n\geq 0$. Its ternary expansion is said to avoid the digit $2$ when no ternary digit equals $2$. Erdős's conjecture. The only p…

Infinitude conjecture for the sets of digit-sum classes. For each $d\geq2$, the sets $\mathcal{S}_d$ and $\mathcal{S}\setminus\mathcal{S}_d$ are infinite. Moreover,…

Infinitude conjecture for primary Carmichael numbers. The following claims are true: the set $4\mathcal{C}'4$ is infinite, and the set $4\mathcal{C}\setminus\mathcal{C}'4$ is infin…

Let $r\ge 1$, let $N_1,\ldots,N_r$ be positive integers, and let $x_1,\ldots,x_r,y_1,\ldots,y_r$ be variables. For each nonnegative integer $n$, write $s_2(n)$ for the sum of the b…

Let $r\ge 1$, let $N_1,\ldots,N_r$ be positive integers, and let $x,y_1,\ldots,y_r$ be variables. For each nonnegative integer $n$, write $s_2(n)$ for the sum of the binary digits…

Let $S_{10}(n)$ denote the sum of the first $n$ decimal digits of one of the numbers $\pi$, $e$, or $\sqrt{2}$. Sub-iterated-logarithm digit-sum conjecture. For each of these three…

Iterated-logarithm conjecture. One should have

Let $p,q$ be positive integers such that $\log p/\log q$ is irrational, and let $S_q(p^n)$ denote the sum of the digits in the canonical base-$q$ representation of $p^n$. Linear av…

Law-of-the-iterated-logarithm conjecture. The normalized digit-sum deviations satisfy

Let $p$ and $q$ be positive integers such that $\log p/\log q$ is irrational. Let $D_q(n,i)$ denote the $i$-th digit in the canonical base-$q$ representation of $n$, and let…

Rational-value conjecture. As $n$ ranges over the positive integers, this ratio will attain every positive rational number. The surrounding discussion identifies this as an expecte…

Let $b,r$ be positive integers with $\gcd(b-1,r)=1$, and let $c$ be any integer. Write $s_b(n)$ for the sum of the base-$b$ digits of $n$. Bounded-offset conjecture. There exists a…

Let $s_2(n)$ denote the sum of the binary digits of the integer $n$, and let $h$ be a fixed integer with $h\geq 2$. Stolarsky's conjecture. … Stolarsky proved the corresponding ass…

Let $m$ be an integer with $(m,3)=1$, and let $k\in(1,m/3)$. Asymptotic comparison conjecture. One has … The conjecture proposes an asymptotic domination relation between digit sum…

Let $k\geq 1$, and let $S_{2k+1,0,2k}(x)$ be the generalized digit sum considered in the paper. Newman-like phenomenon conjecture. One has … and the Newman-like phenomenon becomes…

Let $p$ be a prime, and let $S_{p,0}(n)$ denote the corresponding digit sum. Divisibility conjecture. For every prime $p$, one has … This conjecture accompanies the observed values…