358 problems

Let $Q$ be an indefinite quadratic form in $n\ge3$ variables that is not proportional to a quadratic form with rational coefficients. Oppenheim conjecture. The set … is dense in…

Let $K$ be a totally real number field. A universal ternary classical quadratic form or universal ternary classical quadratic lattice over $K$ represents every totally positive alg…

For a field $k$, say that it is $C_2$ if, for every form $F\in k[X_1,\ldots,X_n]$ of degree $d$ with $n>d^2$, the equation … has a nontrivial solution in $k^n$. Artin's conjecture.…

Positive-definiteness conjecture. Despite the negative signs in the definition of $Q$, the quadratic form $Q$ is positive definite on the cone $C_{\mathbb{R}}$.

Let $k$ be a field of characteristic different from $2$. The fundamental ideal $\mathbf{I}(k)$ is the ideal in the Witt group $\mathbf{W}(k)$ generated by even-dimensional inner pr…

Sun's conjecture. The polynomials $x^2+15y^2+P_5(z)$, $x^2+11y^2+P_5(z)$, $x^2+7y^2+2P_5(z)$, and $4x^2+3y^2+P_5(z)$ should represent every natural number over $\mathbb Z$.

Sun's binomial-cube congruence conjecture. The displayed congruence holds.

Let $f'=[1,3,7,0,1,1]$ be the positive definite ternary quadratic form appearing in line 23 of Kaplansky's list. Kaplansky's conjecture. The form $f'$ represents every positive odd…

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…

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…

Let $d$ be a locally admissible squarefree positive integer, and let … be the graph of the Gauß orthogonal complement, where $\mathcal{S}^2=\mathrm{SO}_3(\mathbb{Z})\backslash S^2$…

Pfister-neighbour conjecture. If $s<\frac{2^{n+1}}{3}$ and $\varphi_1$ is defined over $F$, then $\varphi$ is a Pfister neighbour.

Invariant-lattice classification conjecture. Every primitive, $\operatorname{GL}_3(\mathcal{O}_K)$-invariant lattice $L$ in $\mathcal{V}(\mathcal{O}_K)$ is…

Ohno–Nakagawa reflection conjecture. A reflection theorem holds for pairs of $n$-ary quadratic forms for every odd $n$.

A positive definite integral quadratic form $f$ represents an integer $n$ with $r(n,f)$ denoting the number of such representations. For a square $n^2$, a representation formula is…

Let $F$ be a field. The étale cohomology pointed set $H^1_{{\mathrm{et}}}(F,\mathrm{F}_4)$ parametrizes isomorphism classes of groups of type $\mathrm{F}_4$. Consider the map … ind…

Lang's conjecture. The function field $\mathbb R(X)$ is $C_i$.

Scully's isotropy-index conjecture. One has $k\geq 0$, and

Liouville's conjecture. The number of representations satisfies

Let $T_m=m(m+1)/2$ and let $E(8T_x+8T_y+T_z)$ be the set of positive integers not represented by $8T_x+8T_y+T_z$. Since $8T_x+1$ is an odd square, a sum of two odd squares and a tr…

Oppenheim's conjecture. The equality $m(Q)=0$ holds.

Let $F$ be a field, let $\widetilde{F}/F$ be a field extension of degree $d$, and let $\varphi=N_{\widetilde{F}/F}$ be its norm form. For a field extension $K/F$, say that SNP hold…

Let $w_{ij}$, $1\leq i<j\leq n$, be arbitrary real coefficients, and let $\xi_1,\ldots,\xi_n$ be independent random signs with…

Let $p$ and $q$ be two primes, and let $(1,0,5)$ denote the binary quadratic form $x^2+5y^2$. The primes $p$ and $q$ are each congruent to $3$ or $7$ modulo $20$. Fermat's conjectu…

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…