23 problems

Let $R$ be obtained from the set of positive squares by deleting some of its elements up to $X$ and adding some positive integers, with the total number of changed elements being…

A nontrivial $3\times3$ generalized arithmetic progression of squares is a set of the form … whose entries are squares and which is not degenerate. The cubic square-grid conjecture…

The short square-set energy conjecture. One should have

For a finite set $E$ of squares in the short interval … let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The local energy conjecture. There exists…

The short interval conjecture. For every $\alpha<1$, every such trigonometric polynomial satisfies

For a set $E\subset\{1^2,\ldots,N^2\}$ of squares, let $r_{E+E}(m)$ count representations of $m$ as a sum of two elements of $E$. The logarithmic energy conjecture. There exists a…

For integers $a$ and $b$, consider positive solutions of $x^2\equiv a\pmod b$. The congruence-spacing conjecture. There exists a constant $N$ such that, for every $a$ and $b$, ther…

An affine cube of dimension $d$ in $\mathbb Z$ is a set … where $b_0,\ldots,b_d$ are nonzero integers. Solymosi's conjecture. There exists an integer $d>0$ such that no affine cube…

For a finite set $E$ of squares, write $E+E=\{x+y:x,y\in E\}$. Ruzsa's conjecture. For every $\epsilon>0$, … The source states that Chang's conjecture implies this one, while the B…

For a finite set $E$ of integers, define … and let $r_{E+E}(n)$ be the number of representations of $n$ as a sum of two elements of $E$. Chang's conjecture. For every $\epsilon>0$…

Let $H_3(n)$ denote the quantity used in the paper for the size or counting function of three-dimensional Hilbert cubes of squares. Asymptotic growth conjecture for three-dimension…

Let … be the set of integer squares, and let … be a Hilbert cube of dimension $3$. Existence conjecture for three-dimensional Hilbert cubes of squares. For each…

Let $\mathcal{W}$ be an additive complement of the set of squares, and let $f(n)$ denote the number of representations of $n$ as $w+m^2$ with $w\in\mathcal{W}$ and $m^2$ a square.…

Let $S=\{1^2,2^2,\dots\}$ be the set of square numbers, and let $Q$ be a finite subset of $S$. Write $Q+Q=\{q_1+q_2:q_1,q_2\in Q\}$. Ruzsa's conjecture. There is a positive real nu…

Let $s(X)$ denote the size of the largest subset of $[X]=\{1,\dots,X\}$ containing no two distinct elements whose difference is a square. Sárközy's conjecture. One has … This remai…

Let $s(X)$ denote the size of the largest subset of $[X]=\{1,\dots,X\}$ containing no two distinct elements whose difference is a square. Erdős's conjecture. One has … This conject…

Let $\mathcal{S}$ denote the set of squares, and let $\mathcal{W}=\{w_n\}_{n=1}^{\infty}$ be an additive complement of $\mathcal{S}$, with the elements of $\mathcal{W}$ indexed inc…

Let $B\in\mathbb{N}_{\ge 2}$ and let $D\subseteq\{0,1,\dots,B-1\}$ satisfy $|D|\ge 2$. Let $\alpha_B^D(n)$ be the number of integers $B^{n-\frac12}\le x<B^{n+\frac12}$ such that th…

Let $B\in\mathbb{N}_{\ge 2}$ and let $D\subseteq\{0,1,\dots,B-1\}$ satisfy $|D|\ge 2$. Let $\omega_B^D(n)$ count the relevant square suffixes of length $n$ in base $B$. Suffix-asym…

For a base $B\ge 2$ and a digit set $D\subseteq\{0,1,\dots,B-1\}$ with $|D|\ge 2$, let $\omega_B^D(n)$ denote the number of $n$-digit suffixes in base $B$ from $D$ that occur as th…

Let $\langle a_1',\dots,a_l'\rangle$ be a solution of length $l\ge 2$, where a solution is a digit sequence satisfying the paper's square-digit condition. Prefix-closure conjecture…

Let a pattern be a finite sequence of digits satisfying the square-digit conditions defined in the paper, and let $l$ denote its length. The preceding discussion shows that such pa…

Let $Q(N)$ denote the maximum number of squares in an arithmetic progression of length $N$. A slight refinement of the Erdős–Rudin conjecture states that the maximum is attained by…