Mathematics Problem Archive

If $A$ and $B$ are countably infinite groups, does $A\times B$ have fixed price $1$?...

Given a visibility graph $G$ and a Hamiltonian circuit $C$, determine in polynomial time whether there is a simple polygon whose vertex visibility gra...

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 $L(E_n,1)=0$?...