14 problems

Erdős–Turán conjecture. If $A$ is an asymptotic basis, then the representation function $f_A(n)$ cannot be bounded.

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…

Tang and Yu's conjecture. The equality $R_C(n)=R_D(n)$ cannot hold for all sufficiently large $n$. Chen and Lev disproved the conjecture by constructing a family of partitions of t…

Let $h$ be a positive integer, let $A\subseteq\mathbb{Z}_{\geq 0}$, and let $r_{A,h}(n)$ denote the number of representations of $n$ as a sum of $h$ elements of $A$, with the repre…

Multiplicity conjecture. The number of integers $n\leqslant x$ that are multiply represented satisfies

Let $\mathbb{N}$ be the set of all nonnegative integers, and for $S\subseteq\mathbb{N}$ let $R_S(n)$ denote the number of solutions of $n=s+s'$ with $s,s'\in S$ and $s<s'$. Let…

Erdős's conjecture. There exists an integer $n$ such that the number of solutions of

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…

Let $m\geq 1$ and let … with $1\leq b_0<b_1<\cdots<b_m$. For an infinite subset $\mathcal A$ of positive integers, let $r_{\rm M}(n,\mathcal A)$ denote the number of representation…