10 problems

Cusick's conjecture. For every integer $t\geq 1$,

Asymptotic conjecture. The counting function satisfies

Let $\mathsf s(n)$ be the number of ones in the binary expansion of a nonnegative integer $n$. For $t\geq 0$ and $j\in\mathbb Z$, define … where $\operatorname{dens} A$ denotes the…

Let $b\geq 2$ be an integer, and write the base-$b$ expansion of $\sqrt{2}$ as an infinite digit sequence. For each finite string of digits of length $k$, consider its limiting fre…

Let $k$ be a positive integer. A multiple of a prime $m$ has binary weight $k$ if it can be written as $\sum_{i=1}^k2^{s_i}$ for nonnegative integers $s_i$. The conjecture. For arb…

Let the positive multiples of $3$ be written in binary, and compare, up to a given limit, those having an even number of $1$'s with those having an odd number of $1$'s. Moser's con…

Let $p\neq 2$ be a prime, let $m=2p$, and let $m_lm_{l-1}\cdots m_1m_0$ be the standard base-$2$ representation of $m$. Let $n\geq 1$. Write $n_1$ for the leading digit in the base…

Simple normality conjecture. The number $\sqrt{2}$ is simply normal to base $2$; equivalently, the asymptotic occurrence rate of zeroes in its dyadic expansion is $1/2$, so that

Let $n,r\geqslant 1$. Define $\alpha(N)$ to be the number of $1$'s in the binary expansion of $N$. Binary-weight divisibility conjecture. The sum … is divisible by … The conjecture…

Monotonicity conjecture. For all $n\in\mathbb{N}$, $n\neq 5,6$, the sequence $m_n$ increases monotonically.