Mathematics Problem Archive

Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?...

If the class of atomic models of a complete first order theory is categorical in the $\aleph_n$, is it categorical in every cardinal?...

Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?...

Is the theory of the field of Laurent series over $\mathbb{Z}_p$ decidable? of the field of polynomials over $\mathbb{C}$?...

Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?...

What is the nature of the proof-theoretic ordinal (the smallest ordinal a theory cannot prove well-founded) for second-order arithmetic, ZFC, or stron...

Give a mathematically precise meaning to an exact formula comparable to Onsager's formula for the two-dimensional Ising model, and prove or disprove t...

Find an asymptotic expression for the splitting of the separatrix of a quasiperiodically forced pendulum in the regime where the perturbation series i...

For the Edwards-Anderson Ising spin glass on $\mathbb{Z}^d$ with i.i.d. mean-zero finite-variance nearest-neighbor couplings, prove or disprove the ex...

Give a mathematically rigorous formulation and justification of the nuclear shell model....

Prove that, as the number of nuclei tends to infinity, the ground state of some neutral system of nuclei and electrons approaches a periodic limit, es...

Random point processes corresponding to $\beta$-ensembles, or, equivalently, log gases at inverse temperature $\beta$, are defined for arbitrary $\bet...

The fact that RMT, and the Tracy-Widom distributions, arise in so many problems in so many different areas leads one to the following question: how ca...

Consider the Korteweg--de Vries (KdV) equation u_t + u u_x + u_{xxx} =0 with initial data u(x, t=0) = u_0(x),\qquad x\in \mathbb{R}. In the 1970's, Mc...

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

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...

In 2002, P. Zhou analyzed the behavior of solutions of the Cauchy problem for perturbations of the defocusing NLS equation i u_t + u_{xx}- 2|u|^2 u - ...

Give a rigorous explicit solution of the fixed-frequency scalar-wave diffraction problem for two finite slits in the plane, including asymptotics of t...

Determine whether there exist bound entangled bipartite quantum states with negative partial transpose....

Is discrete logarithm modulo a prime randomized polynomial-time reducible to computing $g^{xy}$ from $g,g^x,g^y$?...

Is there an absolute upper bound for the size of a rational Diophantine $m$-tuple, a set of nonzero rationals for which the product of every two disti...

Let $U$ be the set of integers $n\not\equiv2\pmod4$ for which there are at most two distinct $D(n)$-quadruples. Is $U$ finite?...

Are there infinitely many Diophantine triples that are also $D(n)$-triples for three distinct integers $n\ne1$?...

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 $P_n$ be the supremum of the sizes of nondegenerate polynomial $D(n)$-tuples over $\mathbb{Z}[X]$. Find an upper bound for $P_n$ depending only on...

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 $m=n$...

For any $\varepsilon>0$, there is a constant $C(\varepsilon)>0$ such that, for any positive integers $x$, $y$, $p$, $q$ satisfying $x^p\not= y^q$, the...

If $x$ and $y$ are positive integers with $y^2\not=x^3$, then $$ |y^2-x^3|\ge C\max\{y^2,x^3\}^{1/6}. $$...

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...

Let $\lambda_1,\ldots, \lambda_n$ be $\mathbb{Q}$-linearly independent complex numbers. Assume that the numbers $e^{\lambda_1},\ldots,e^{\lambda_n}$ a...

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 $d_0=1$, $d_1=0$, $d_2=1$....

The numbers $\pi$, $\zeta(3),\zeta(5),\ldots,\zeta(2n+1),\ldots$ are algebraically independent over $\mathbb{Q}$....

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

There exists a positive absolute constant $c$ such that, for any nonzero algebraic number $\alpha$ which is not a root of unity, $$ \mathrm{M}(\alpha)...

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 ...

Let $x_1,\ldots,x_n$ be $\mathbb{Q}$-linearly independent complex numbers. Assume that for any $\varepsilon>0$, there exists a positive number $H_0$ s...

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 $n$ be a positive integer. For almost all $n$-tuples $(x_1,\ldots,x_n)$, there are positive constants $c$ and $D_0$ (depending on $n$, $x_1,\ldots...

Is computation of a curve's zeta function polynomial simultaneously in the genus $g$ and in $\log q$?...

For a prime $p$, conjecturally $p$ does not divide the class number of the maximal real subfield $\mathbb{Q}(\zeta_p+\overline{\zeta_p})$ of the $p$th...

Let $\zeta_p:\mathbb{Z}_p\to\mathbb{Q}_p$ be the $p$-adic zeta function. Is $\zeta_p(k)\ne0$ for every even integer $k$?...

Given an integer $n$, determine whether there are rational numbers $x,y,z$ satisfying $x^2+y^2=z^2$ and $xy=2n$; equivalently, determine whether $n$ i...

Is every integer $n\equiv5,6,$ or $7\pmod 8$ a congruent number?...

For $E_n:y^2=x^3-n^2x$, is $n$ a congruent number if and only if $L(E_n,1)=0$?...

Find the two prime factors of the RSA-1024 challenge integer $1350664108659952233496032162788059699388814756056670275244851438515265106048595338339402...