4 problems

Generalized nonintegrality conjecture. There is a positive integer $N$ such that for every integer $n\geq N$ and every integer $k$ with $1\leq k\leq n$, neither…

Let $f(x)$ be a nonzero polynomial of integer coefficients. Let $Z_f:=\{x\in\mathbb{Z}:f(x)=0\}$, let $\{a_k\}_{k=1}^{\infty}:=\mathbb{Z}^+\setminus Z_f$ be arranged in increasing…

Let $f_n(s,t)$ be the generalized Fibonacci sequence, let $\mathbb{Z}^+$ denote the positive integers, and let $\lfloor\cdot\rfloor$ be the floor function. ACMS conjecture. If…

Let $f_n(s,t)$ be the generalized Fibonacci sequence, and consider reciprocal sums over positive integers $n$ and $r$. The preceding theorem gives floor formulas for sums involving…