Mathematics Problem Archive

Is the open mapping theorem for separable Banach spaces provable in $\mathsf{RCA}_0$, or at least in $\mathsf{WKL}_0$?...

Determine the exact reverse-mathematical strength of the Krein–Šmulian theorem for separable Banach spaces....

Is Szemerédi's theorem provable in $\mathsf{ACA}_0$? More generally, determine its reverse-mathematical strength....

For the two-dimensional square-lattice model of independent electrons at density $1/4$, does magnetic flux $\pi/2$ per plaquette minimize the ground-s...

Prove that Bose-Einstein condensation occurs in a continuum model of a weakly interacting Bose gas, or determine whether the long-held assertion fails...

Prove that the Anderson model has purely absolutely continuous spectrum in dimension $\nu\geq 3$, for suitable disorder width $b-a$, in some energy ra...

Prove that the spectrum of the Anderson model in dimension $\nu=2$ is dense pure point....

For the Anderson model in dimension $\nu\geq3$ and disorder strengths $|b-a|$ admitting absolutely continuous spectrum, prove that $\sum_{n\in\mathbb ...

For the $N$-electron Coulomb ground-state energy $E(N,Z)$, determine the asymptotics of the ionization energy $\delta E(Z)=E(Z,Z-1)-E(Z,Z)$ as $Z\to\i...

What I have in mind here is not a specific problem, but a project, a very large scale project. The six (nonlinear) Painlev\'e equations form the core ...

Construct a set of at least four mutually unbiased bases in dimension six, or prove that there are no seven mutually unbiased bases in $\mathcal{H}_6$...

Given $n\in\mathbb{N}$, can a prime $p>n$ be found in deterministic polynomial time?...

Given coprime $a,n\in\mathbb{N}$, can a prime $p\equiv a\pmod n$ be found in deterministic polynomial time?...

Is complete integer factorization $C_5$ in deterministic polynomial time $P$?...

Is complete integer factorization $C_5$ in randomized polynomial time $R$?...

Does there exist a set $S\subset\mathbb{N}$ of positive lower asymptotic density for which complete factorization of every input $n\in S$ is in determ...

Given $n$, can one find $r,s\in\mathbb{N}$ with $n=r^2s$ and $s$ squarefree in deterministic polynomial time?...

Is complete integer factorization randomized polynomial-time reducible to computation of the squarefree part?...

Can $\omega(n)$, the number of distinct prime factors of $n$, be computed in deterministic polynomial time?...

Can one decide in deterministic polynomial time whether a coprime integer $a$ is a square modulo a composite $n$?...

Is complete integer factorization randomized polynomial-time reducible to deciding quadratic residuosity modulo a composite?...

Given a prime $p$, can a quadratic nonresidue modulo $p$ be found in deterministic polynomial time?...

Given a sign vector $\varepsilon\in\{-1,1\}^k$, can the least prime $p$ satisfying $(p_i/p)=\varepsilon_i$ for every $i\le k$ be found in deterministi...

Given a prime $p$ and a quadratic residue $a$, can a square root $x^2\equiv a\pmod p$ be found in deterministic polynomial time?...

Given a prime $p$ and $f\in(\mathbb{Z}/p\mathbb{Z})[x]$ known to have a root, can a root be found in deterministic polynomial time?...

Given a prime $p$ and $f\in(\mathbb{Z}/p\mathbb{Z})[x]$, can the complete irreducible factorization of $f$ be found in deterministic polynomial time?...

Given a prime $p$ and $b$, can one decide in deterministic polynomial time whether $b$ generates $(\mathbb{Z}/p\mathbb{Z})^*$?...

Is recognition of primitive roots modulo a prime in randomized polynomial time $R$?...

Given a prime $p$, can a generator of $(\mathbb{Z}/p\mathbb{Z})^*$ be found in deterministic polynomial time?...

Given a prime $p$ and $a$ coprime to $p$, can $\operatorname{ord}_p(a)$ be computed in deterministic polynomial time?...

Given a prime $p$ and elements $g,b$ with $b$ in the subgroup generated by $g$, can an exponent $x$ satisfying $g^x\equiv b\pmod p$ be found in determ...

Given $g,b,n$ such that $g^x\equiv b\pmod n$ has a solution, can such an exponent $x$ be found in deterministic polynomial time?...

Given $k,m,n$ with odd $n$ and $\gcd(km,n)=1$, can integers $x,y$ satisfying $x^2-ky^2\equiv m\pmod n$ be found in deterministic polynomial time?...

Given a prime $p$ and $n$, can one construct in deterministic polynomial time an elliptic curve over $\mathbb{F}_p$ having exactly $n$ points whenever...

Given an elliptic curve over $\mathbb{F}_p$ and points $P,Q$ such that $P=nQ$ for some $n$, can such an $n$ be found in deterministic polynomial time?...

Can one decide in deterministic polynomial time whether $x^2-dy^2=-1$ has an integral solution?...

For each $n\in\{-3,3,5,8,12,20\}$, prove that no set of four distinct positive integers has property $D(n)$, meaning that every pairwise product plus ...

For every nonzero integer $n$ that is not a square, are there only finitely many $D(n)$-quadruples?...

Does there exist a set of four nonzero rational numbers $\{a_1,a_2,a_3,a_4\}$ such that $a_i a_j+1$ is a rational square for every $1\le i,j\le4$, inc...

Let $f\in\mathbb{Z}[X,Y]$ be a polynomial such that the equation $f(x,y)=0$ has only finitely many solutions $(x,y)\in \mathbb{Z}\times\mathbb{Z}$. Gi...

There exists a positive number $C$ which depends only on $L$ and $H$ with the following property. Let $m$, $x$ and $y$ be rational integers with $m\ge...

Let $k\ge 2$ be an integer and $\alpha_1,\ldots,\alpha_n$ be non-zero elements in a field $K$ of zero characteristic, such that no quotient $\alpha_i/...

If there is no prime in the interval $[n+1,n+k]$, then the product $(n+1)\cdots(n+k)$ has at least $k$ distinct prime divisors....

Given an increasing sequence $n_1<n_2<\cdots<n_k$ of positive integers such that $n_1,n_2,\ldots,n_k$ are multiplicatively dependent, there exists a p...

Fix a positive integer $m$ for which the equation $$ m^2 + m_1^2 + m_2^2 = 3 mm_1m_2 $$ has a solution in positive integers $(m_1,m_2)$ with $0<m_1\le...

There exist real numbers $\varepsilon$, $\alpha$ and $\beta$ with $0<\varepsilon<1/2$, $\alpha\ge 1$ and $\beta\ge 0$, and a positive integer $B$, suc...

For any $\varepsilon>0$, there exists a constant $C(\varepsilon)>0$ such that, for any nonzero rational integers $a_1,\ldots,a_m$, $b_1,\ldots,b_m$ wi...

Let $\theta$ be real algebraic number of degree at least $3$. Then inequality (2.11) has infinitely many solutions in integers $p$ and $q$ with $q>0$ ...

There exists an absolute constant $c>0$ such that $$ \Vert \log a\Vert>a^{-c} $$ for all integers $a\ge 2$....

Let $k$ be a positive integer, $y_1,\ldots,y_k$ complex numbers which are linearly independent over $\mathbb{Q}$, $\alpha_1,\ldots,\alpha_k$ nonzero c...