7 problems

Let $A\subseteq\mathbb N$ and let $h\geq 1$ be an integer. Define the ordered $h$-fold representation function by … The complement $\mathbb N\setminus A$ is co-infinite when it is…

Sárközy–Sós conjecture. There exists some infinite set of positive integers $\mathcal{A}$ such that $r_{\mathcal{A}}(n;k_1,\dots,k_d)$ is constant for $n$ large enough if and only…

Chen–Lev strengthened conjecture. If $R_C(n)=R_D(n)$ for every positive integer $n$, then there exists an integer $l\ge1$ such that

Chen–Lev conjecture. If $R_C(n)=R_D(n)$ for every positive integer $n$, then there exists an integer $l\ge1$ such that

Chen–Lev conjecture. If $R_C(n)=R_D(n)$ for every positive integer $n$, then there exist positive integers $d_{i_1}<d_{i_2}<\cdots$ belonging to $D$ such that

Kiss–Rozgonyi–Sándor conjecture. The equality $R_{h,C}^{(1)}(n)=R_{h,D}^{(1)}(n)$ from some point onward holds if and only if there exist positive integers $n_0$ and $M$, and finit…

Let $\mathcal{A} = \{a_{1}, a_{2},\ldots\}$ be a set of nonnegative integers satisfying $a_n \leq cn^2$ for all $n$, where $c>0$ is a real constant. Let $R_A(n)$ denote the number…