142 problems

Let $m$ be a positive integer with $m\equiv 1\pmod{24}$. Suppose that all prime divisors of $m$ are congruent to $1$, $5$, $7$, or $11$ modulo $24$, and write … where…

Classification conjecture. Then $\varphi$ is a Pfister neighbour.

Density conjecture. For every $\epsilon>0$, there exists some positive integer $d$ such that if $I\subseteq\mathbb{Z}^2$ has upper density greater than $\epsilon$, then there is a…

Failure of the local-global principle. The local-global principle for isotropy of quadratic forms fails to hold in dimension $2^n$ over $K$.

Let $\mathcal{P}$ be an integral circle packing, let $\mathcal{K}\subset\mathbb{Z}$ be its set of curvatures, and let $\mathcal{K}_a$ be the set of integers passing all local obstr…

Let a cycle of reduced forms have caliber $l$, discriminant $a^2+4$, and sum $n$. Regularity conjecture. There is an integer $r\geq 0$ such that … If $r>0$, the cycle consists of i…

Finite test set conjecture. A sum of triangular numbers $f$ represents all odd integers if and only if it represents the integers

Let $q$ be an anisotropic quadratic form over a field $k$. Its first Witt index is the Witt index of $q$ over the function field of its associated projective quadric. Converse to K…

Converse to Karpenko's theorem. If the first Witt index of $q$ is greater than $1$, then $q$ is ruled over $k$.

Let ${\mathcal{V}}$ be an $n$-dimensional, non-degenerate quadratic space, let $[n-1]=\{1,\ldots,n-1\}$, let ${\mathcal{C}}_n$ be the subgroup of chessboard elements in the symmetr…

A perfect form is a positive definite quadratic form whose associated Delone subdivision is determined by its minimal vectors; Voronoi's first perfect form is the first form in Vor…

Let $f:\mathbb{R}^2\to\mathbb{R}$ be an integer quadratic form, meaning that its coefficients are integers. The form $f$ has semigroup property if the product of any two values rep…

Let $m\geq 3$ be odd, let $n$ be a non-negative integer, and let $f$ be a quadratic polynomial in $n$ variables modulo $m$. Write $q=2\cos\left(\frac{\pi}{2m}\right)$ and let…

Let $m\geq 3$ be odd, let $n$ be a non-negative integer, and let $f$ be a quadratic polynomial in $n$ variables modulo $m$. Write $q=2\cos\left(\frac{\pi}{2m}\right)$ and let…

Shapiro's conjecture. $(A,\sigma)$ admits a decomposition as a tensor product of $r$ quaternion algebras with involution in which each quaternion algebra is split.

Let $Q^{c1}_6$ be the positive quadratic form defining the lattice covering studied in the paper, and let lattice coverings in dimension $6$ be compared by their covering density.…

Vishik's conjecture. There is some $i\in\{1,2,\dots,n+1\}$ such that

Reciprocal counting-function conjecture. For every $x$, one has

Weighted counting-function conjecture. For $x\geq 8$, one has

Schmutz Schaller's counting-function conjecture. For every $x$, one has

For each odd square-free integer $t$, let $\Delta(t)=r_{Q_1}(t)-r_{Q_2}(t)$, and define … … and $N_{\ne 0}(X)=N_+(X)+N_-(X)$. Sign equidistribution conjecture. As $X\to\infty$, … E…

Bernoulli–quadratic-form conjecture. For every prime $p$,

Let $T_o(q)=\sum_{n\geq 0}t_o(n)q^n$, and let $n$ be a nonnegative integer. Parity conjecture. If $8n+9$ has a prime divisor $p\equiv 5,7\pmod{8}$ to odd exponent, equivalently if…

Let $\mathbb{N}$ denote the nonnegative integers, and let $\mathcal{C}_0$ be the set of core abaci of affine type $C_3^{(1)}$ with charge $0$. The preceding proposition identifies…

Let $q:\mathbb{Z}^m\to\mathbb{Z}$ and $Q:\mathbb{Z}^n\to\mathbb{Z}$ be integral quadratic forms with $m<n$, and let … Assume that $q$ is primitively represented by an element of th…