318 problems

I conjectured long ago that if … then the $a_i$'s contain arbitrarily long arithmetic progressions. If true this of course implies that there are arbitrarily long arithmetic progre…

Rudin's conjecture. The maximal number of squares satisfies

Let $[n]=\{1,\ldots,n\}$, and let $\mathcal{A}$ be a family of subsets of $[n]$ or a family of labelled graphs on $n$ vertices. In the first setting, require that every pair of set…

Let $b$ and $l$ be coprime integers, and consider the first prime $p\equiv b\pmod l$. Granville–Pomerance conjecture. For infinitely many choices of $l$, this first prime satisfies…

Let $k$ be the length of the progression, let $d$ be its common difference, and let $A_{k,d}$ be the adjustment factor defined by the local prime-divisibility probabilities. Prime…

Let $\mathcal{A}$ be the set of positive integers $N$ for which the corresponding triple $N,N+d,N+2d$ is a three-term arithmetic progression of consecutive powerful numbers. Partit…

Lemke Oliver–Soundararajan conjecture.

Sharp upper-bound conjecture. For all $d \geq 2^{2|v|+1}$,

A Mordell elliptic curve is a curve with Weierstrass form … An $x$- or $y$-arithmetic progression on $E_k$ is an arithmetic progression in the corresponding coordinate. Mohanty's c…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with a specified Weierstrass equation. Consider arithmetic progressions contained in the set of $x$-coordinates of points in…

Let $\mathbb{P}(x)$ be the set of prime numbers not exceeding $x$, and let $M_{\mathbb{P}(n)}(n)$ denote the corresponding prime-restricted maximum packing number. Prime-restricted…

Keating–Rodgers–Roditty-Gershon–Rudnick's variance conjecture. When $Q^{1+\varepsilon}<X<Q^{k-\varepsilon}$ and $X\to\infty$,

Barban–Davenport–Halberstam conjecture. Under these conditions,

Let $x\geq2$, $q\in\mathbb{N}$, and $a,h\in\mathbb{Z}$ with $(a,q)=1$. Define … where $\mathbf{1}_{(a(a+h),q)=1}$ equals $1$ when both $a$ and $a+h$ are coprime to $q$, and equals…

Let $A>0$ be a fixed real number, and let $y$ and $q$ be large positive numbers satisfying $q\leq y^A$. For $x$ tending to infinity with … let $\Psi(x,y;q,a)$ count the $y$-smooth…

Graham's square conjecture. The set $S$ must contain an axis-aligned square.

Let $S(A)=(a_n)$ be a Stanley sequence, meaning the sequence generated greedily from a finite $3$-free set $A\subset\mathbb N_0$. Rolnick's growth-rate conjecture. For all $n$ larg…

Let $n$ be a positive integer. A three-term arithmetic progression $(a,a+r,a+2r)$ in $\mathbb{Z}/n\mathbb{Z}$, with not all terms equal, is preserved by a permutation…

Threshold conjecture. For every fixed positive integer $d$, there should be a threshold around

For every even integer $k\ge 4$, let $\rho_k(\alpha)$ denote the minimum asymptotic density of $k$-term arithmetic progressions in a $U^{k-2}$-uniform set of density $\alpha$ (with…

Jungić–Licht–Mahdian–Nešetřil–Radoičić conjecture. The quantity $T_k$ satisfies

Roth's conjecture. For every such function $f$, the lower bound $D(f)\geqslant cn^{1/4}$ should be improvable to $D(f)\geqslant n^{1/2-o(1)}$. Roth's theorem gives the stated…

The Thue–Morse odd-difference bound conjecture.

Let $q$ be a sufficiently large positive integer and let $a$ be an integer with $(a,q)=1$. Erdős–Odlyzko–Sárközy conjecture. There exist primes $p_1,p_2\leq q$ such that … This con…

Martin's conjecture. The set