Mathematics Problem Archive

In 1999, J. All these early examples were ``integrable'' in the sense that certain powerful algebraic tools, e.g., determinantal particle systems, the...

Standard algorithms to compute the eigenvalues of a random matrix $H$ are completely integrable Hamiltonian systems (see ). The question raised in was...

Initial boundary value problems for integrable systems in $1+1$ dimensions are of great interest. It turns out, however, that in Fokas' approach the g...

Solutions of the Cauchy problem for linear dispersive equations can be expressed in terms of Fourier integral operators. This is a very challenging pr...

Can one decide in deterministic polynomial time whether an integer $n$ is squarefree?...

Let $C_{10}$ find $x$ satisfying $x^e\equiv a\pmod n$ under $\gcd(e,\varphi(n))=\gcd(a,n)=1$. Is complete integer factorization randomized polynomial-...

Given a prime $p$ and degree $d$, can an irreducible polynomial of degree $d$ over $\mathbb{F}_p$ be constructed in deterministic polynomial time?...

Is complete integer factorization deterministically polynomial-time reducible to discrete logarithms modulo composites?...

Is complete integer factorization deterministically polynomial-time reducible to computing $\varphi(n)$?...

Given $a,b$ and a prime $p$ with nonsingular curve $y^2=x^3+ax+b$, can a point on the curve modulo $p$ be found in deterministic polynomial time?...

Does there exist a constant $c$ for which one can find in deterministic polynomial time a nonzero lattice vector of length at most $n^c$ times the min...

Given $f\in\mathbb{Q}[x]$, can the degree of its splitting field, equivalently the order of its Galois group, be computed in deterministic polynomial ...

Given $d\in\mathbb{N}$, can the class number $h(-d)$ of binary quadratic forms of discriminant $-d$ be computed in deterministic polynomial time?...

Can $\gcd(a,b)$ be computed in the parallel complexity class $NC$?...

Can two positive integers $a,b$ be multiplied using $O(\log(ab))$ bit operations?...

Determine the least positive integer $n_1$ and the greatest negative integer $n_2$ for which a $D(n_i)$-quintuple exists....

Does there exist a rational Diophantine septuple, that is, seven nonzero rational numbers whose pairwise products plus $1$ are rational squares?...

For which rational numbers $q$ do there exist infinitely many rational $D(q)$-quintuples?...

Let $k$ be a positive integer. The equation $$ x^p-y^q=k, $$ where the unknowns $x$, $y$, $p$ and $q$ take integer values, all $\ge 2$, has only finit...

There exists a positive integer $k$ such that, for $m$ and $n$ positive integers, the conditions $$ R(m+i)=R(n+i)\quad (i=0,\ldots,k-1) $$ imply $m=n$...

If $a$ and $b$ are two positive integers with $a<b$ and $R(a)=R(b)$ then there is a prime $p$ with $a< p< b$....

Let $(\varepsilon_n)_{n\ge 0}$ be a sequence of elements in $\{0,1\}$. Assume that the real number $$ \sum_{n\ge 0}\varepsilon_n 3^{-n} $$ is irration...

As a $\mathbb{Q}$-algebra, $\mathfrak{Z}$ is the direct sum of $\mathfrak{Z}_p$ for $p\ge 0$....

For $p\ge 3$ we have $$ d_p=d_{p-2}+d_{p-3} $$ with $d_0=1$, $d_1=0$, $d_2=1$....

$\overline{G}$ is a universal odd distribution with values in groups where multiplication by $2$ is invertible....

Let $\tau\in\mathbb{C}$ have positive imaginary part. Assume that $\tau$ is not quadratic. Set $q=e^{2i\pi\tau}$. Then at least $4$ of the $5$ numbers...

For each integer $n\ge 1$ there exists a positive constant $c(n)$ such that, for any algebraic subvariety $V$ of $\mathbb{G}_m^n$ which is defined ove...

There exists a positive constant $c$, depending only on $A$ and $\mathcal{L}$, such that for any $P\in A(\overline{\mathbb{Q}})$ which has infinite or...

For any positive integer $n$ and any real number $\theta$ which is either transcendental or else is algebraic of degree $>n$, there exists a positive ...

Assume that $K=\mathbb{Q}$ and that $V(\mathbb{Q})$ is Zariski dense; is $Z$ a union of connected components of $V(\mathbb{R})$?...

Let $A$ be a simple Abelian variety of dimension $g$ over a number field $K$ embedded in $\mathbb{R}$. Denote by $\ell$ the rank over $\mathbb{Z}$ of ...

Let $n \ge 2$. Denote by $\mathrm{ZH}_n$ the set of real numbers $\xi$ with the following property: there exists $c_1(\xi) >0$ and $c_2(\xi)>0$ such t...

Let $(n_i)_{i\ge 0}$ be an increasing sequence of positive integers. Assume there is a prime number $p$ such that the power series $$ \sum_{i\ge 0}z^{...

Given an elliptic curve $E/\mathbb{F}_q$ and $P\in E(\mathbb{F}_q)$, characterize maps or isogenies $\psi:E\to E'$ for which $\psi(P)$ can be evaluate...

Find bilinear structures that are useful for cryptographic constructions....

Find orders $\mathcal{O}$ whose Picard groups contain $\mathbb{Z}/\ell$, admit compact element representations, and allow group composition in $O(\log...

Make the proposed realization of existing finite-field discrete-logarithm systems inside class groups explicit for practical systems....

Determine whether Weil descent compromises the security of the proposed class-group realization of finite-field discrete-logarithm systems....

Explicitly construct global algebras or Brauer classes with prescribed local data, especially when the local splitting fields are dihedral....

Determine whether natural de Rham cup-product pairings can be used to improve zeta-function computations....

Remove the imaginary-hyperelliptic and $p\ne2$ restrictions from the complexity bound stated in the slides....

Test and analyze whether deleting extra points and fixing the lift $x\mapsto x^p$ improves Frobenius lifts for nondegenerate curves....

Develop the higher-dimensional analogue of the nondegenerate-curve Frobenius-lift method....

Find useful deformations of nondegenerate curves together with easy starting matrices for Frobenius computation....

Work out effective zeta-function computations for nondegenerate surfaces in toric threefolds....

Factor $N=p^rq^s$ with $r\approx s$ using fewer known bits of the factors than existing methods require....

Develop methods to factor an integer when the known bits of its factors are nonconsecutive....

For large $N$, efficiently find the required short five-term geometric progressions modulo $N$ that avoid first- and second-order recurrence, thereby ...

Study the distribution of group structures $E(\mathbb{F}_q)$ as elliptic curves $E/\mathbb{F}_q$ vary; in particular, determine the correct nonuniform...

Is the exponent $e_q(E)$ of $E(\mathbb{F}_q)$ typically close to $q$?...