1,652 problems

Erdős–Heilbronn conjecture. For every nonempty subset $A\subseteq {\mathbb Z}/p{\mathbb Z}$,

Let $A\subset\mathbb{F}_{2}^{n}$ be a set with $|A+A|\leqslant K|A|$. Polynomial Freiman-Ruzsa conjecture in characteristic two. Then $A$ is covered by at most $2K^{C}$ cosets of s…

Sárközy's conjecture. For all sufficiently large primes $p$, the quadratic residues modulo $p$ are additively irreducible.

Let $m$ be a positive integer, let $h$ be as in Theorem 8, and let $\mathcal{C}_{(2,2^m+1,3,h)}$ denote the BCH code in that theorem. Its minimum distance is denoted by $d$. Minimu…

Let $\mathbb{F}$ be a finite field. An excluded minor for a class of matroids is a matroid not in the class whose proper minors all belong to the class. Rota's conjecture. The clas…

Let $X$ be a smooth projective variety over a finite field, let $i,n$ be integers, and let $\ell$ be a prime invertible in the finite field. Parshin's conjecture. The motivic cohom…

Weil's conjecture. The rational function $Z(X_p,T)$ has the form

Let $n$ be a positive integer, let $\lambda,\mu\vdash n$, and let $q$ be a prime power. Write $\operatorname{GL}_n(\mathbb F_q)$ for the general linear group and…

Let $\mathbb{F}_q$ be a finite field and let $n\geq 1$. Write $\zeta(\mathbb{F}_q,s)$ for the zeta function of $\mathbb{F}_q$, and let $\mathrm{K}_i(\mathbb{F}_q)$ denote its algeb…

Let $m>1$ be an integer and let $p$ be an odd prime. Define … where $x$ denotes the fractional part of a real number $x$. Zhi-Wei Sun's conjecture. For the monomial…

An irrational algebraic number has a base-$b$ expansion that is normal for every integer base $b\geq 2$. Borel's normality conjecture. Every irrational algebraic number is normal.…

Erdős–Falconer distance conjecture. If $|E|\gg q^m$, then $|\Delta_Q(E)|\gg q$.

Let $\mathcal{C}$ be an $[n,k]$ MDS code over the finite field $\mathbb{F}_q$. Segre's MDS conjecture. Then … This conjecture concerns the maximum possible length of MDS codes over…

Let $G$ be a connected graph, let $X_G$ be its graph hypersurface, and write $[X_G]_q=|X_G(\mathbb F_q)|$ for its point-counting function over finite fields. Inspired by the appear…

Let $A\subseteq\mathbb{F}_2^n$ have doubling constant $K$, meaning that $|A+A|\leq K|A|$. A Freiman–Ruzsa theorem says that $A$ can be covered by $F(K)$ translates of a subspace…

Let $q\geq 3$ be a prime power and let $n$ be a positive integer. Consider monic polynomials of degree $n$ in $\mathbb{F}_q[x]$, whose coefficients include the constant term throug…

Let $\mathbb{F}_q$ be a finite field of characteristic $p$, let $\ell$ and $\ell'$ be distinct primes with $\ell,\ell'\neq p$, and let $X$ be a normal variety over $\mathbb{F}_q$.…

Let $q=p^{\ell}$ be a prime power, let $m$ be a positive integer, and let $N$ divide $(q^m-1)/(q-1)$. Let $C_0^{(N,q^m)}$ be the subgroup of $\mathbb{F}_{q^m}^*$ of index $N$, and…

Wolff's finite-field Kakeya conjecture. There is a constant $c_n>0$, depending only on $n$, such that every Kakeya set $K\subset\mathbb{F}_q^n$ satisfies

Finite-field Kakeya conjecture. Such a Kakeya set should have cardinality comparable to that of the ambient space, namely $|K|\approx|\mathbb{F}_q^n|=q^n$ up to a multiplicative co…

Let $d\geq 3$, let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements, and let a weak Nikodym set be a subset of $\mathbb{F}_q^d$ containi…

Let $G$ be a connected graph, let $cs0(G;\mathbf{x})$ be the spanning-tree sum … where $\mathcal{T}(G)$ is the set of spanning trees of $G$ and $\mathbf{x}\in\mathbb{F}_q^{E(G)}$ a…

A function $f$ over finite fields of characteristic $2$ is exceptional APN if it is APN over infinitely many extensions of the base field. The Gold functions are $f(x)=x^{2^k+1}$,…

Helleseth's vanishing conjecture. If $q>2$ and $s\equiv 1 \pmod{p-1}$, then $s$ is singular.

An irreducible cyclic code is denoted by $\mathcal{C}(k,q)$, and a code is 2-weight when its nonzero codewords have exactly two distinct Hamming weights. The semiprimitive codes, s…