Mathematics Problem Archive

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

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?...

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...

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 ...

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?...

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

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

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 $P_n$ 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 $f\in\mathbb{Z}[X,Y]$ 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...

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 $\theta$ 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...

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...

Let $x_1,x_2$ be two $\overline{\mathbb{Q}}$-linearly independent complex numbers and $y_1,y_2$ be also two $\overline{\mathbb{Q}}$-linearly independe...

Let $x_1, x_2$ be two $\mathbb{Q}$-linearly independent complex numbers and $y_1, y_2$ be also two $\mathbb{Q}$-linearly independent complex numbers. ...

For any $4\times 4$ skew-symmetric matrix $\mathrm{M}$ with entries in $\mathcal{L}$ and rank $\le 2$, either the rows of $\mathrm{M}$ are linearly de...

The two numbers $$ \log\alpha\quad\text{and}\quad \alpha^\beta $$ are algebraically independent over $\mathbb{Q}$....

The $d-1$ numbers $$ \alpha^\beta,\; \alpha^{\beta^2},\ldots, \alpha^{\beta^{d-1}} $$ are algebraically independent over $\mathbb{Q}$....

The $d$ numbers $$ \log\alpha,\; \alpha^\beta,\; \alpha^{\beta^2},\ldots, \alpha^{\beta^{d-1}} $$ are algebraically independent over $\mathbb{Q}$....

Let $\beta_1,\ldots,\beta_n$ be $p$-adic algebraic numbers in the domain of convergence of the $p$-adic exponential function $\exp_p$. Then the $n$ nu...

Let $\alpha$ be a non-zero algebraic number in the domain of convergence of the $p$-adic logarithm $\log_p$, and let $\beta$ be a $p$-adic cubic algeb...

Given an absolute constant $c$ and polynomials $P_1,\ldots,P_m$ with a total of $N$ coefficients and no common complex zeros, there is no program to f...

Let $\Sigma$ be a finite subset of $\mathbb{C}^n$ and $\varepsilon$ a positive number. There exists a positive number $r_0(\Sigma,\varepsilon)$ such t...

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

At least three of the four numbers $$ \pi,\; \Gamma(1/5),\; \Gamma(2/5), \; e^{\pi\sqrt 5} $$ are algebraically independent over $\mathbb{Q}$....

Let $\mathcal{E}_1,\ldots,\mathcal{E}_n$ be pairwise non isogeneous elliptic curves with modular invariants $j(\mathcal{E}_h)$. For $h=1,\ldots,n$, le...

Given an elliptic curve with Weierstrass equation $y^2=4x^3-g_2x-g_3$, a nonzero period $\omega$, the associated quasi-period $\eta$ of the zeta funct...

Let $q_1,\ldots,q_n$ be nonzero algebraic numbers in the unit open disc such that the $3n$ numbers $$ J(q_i), \; DJ(q_i),\; D^2J(q_i)\qquad (i=1,\ldot...

Let $q_1$ and $q_2$ be two nonzero algebraic numbers in the unit open disc. Suppose that there is an irreducible element $P\in\mathbb{Q}[X,Y]$ such th...

Is there such a bound depending polynomially on the degree and height of $P$?...

Are there entire transcendental functions $f(z)$ such that if $x$ is a Liouville number then so is $f(x)$?...

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 each positive integer $n\ge 1$ there exists a positive number $c(n)$ having the following property. Let $\alpha_1,\ldots,\alpha_n$ be multiplicati...

For each positive integer $n\ge 1$ there exists a positive number $c(n)$ such that, if $\underline{\alpha}=(\alpha_1,\ldots,\alpha_n)$ is a $n$-tuple ...

For $\theta\in(0,\pi)$, define $$ V_\theta=\{re^{it}\; ;\; r>0,\; |t|>\theta\}. $$ Compute $L(V_\theta)$ in terms of $\theta$....

Let $\underline{\theta}=(\theta_1,\ldots,\theta_m)$ be a $m$-tuple of complex numbers. Define $$ t=\operatorname{trdeg} \mathbb{Q}(\underline{\theta})...

Let $\theta\in\mathbb{C}^m$. There is a positive constant $c$, depending only on $\theta$ and $m$, with the following property. Let $k$ be an integer ...

There exist two positive absolute constants $c_1$ and $c_2$ with the following property. Let $\lambda_1,\ldots,\lambda_m$ be logarithms of algebraic n...