4 problems

Cusick's simplified conjecture. For all $t\geq 0$, one has $c_t+c_{t'}>1$; equivalently, at least one of $t$ or $t'$ satisfies Cusick's conjecture. This is presented as an easier s…

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…

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.