Mathematics Problem Archive

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...

There exists a positive absolute constant $C$ with the following property. Let $\alpha_1,\ldots,\alpha_n$ be nonzero algebraic numbers and $\log\alpha...

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...

Let $A$ be a simple abelian variety over $\mathbb{Q}$, $\exp_A:\mathbb{R}^g\rightarrow A(\mathbb{R})^0$ the exponential map of the Lie group $A(\mathb...

Let $m$, $n$, $k$ be positive integers and $a_{ij\kappa}$ rational integers ($1\le i\le n$, $1\le j\le m$, $1\le\kappa\le k$). For $\underline{x}=(x_1...

For any $\varepsilon>0$ there exists $S_0>0$ (depending on $\varepsilon$, $\gamma_1,\ldots,\gamma_m$ and $\mathcal{K}$) such that, for any $S\ge S_0$ ...

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...

Turn $H^2(G_K,K_s^*)$ into an explicitly computational group....

Explicitly describe the cocycle $c_u$, equivalently fast-compute invariants of local algebras split by the generalized-dihedral extensions specified i...

Adapt Schoof's method to compute zeta functions of curves without unfavorable dependence on the genus....

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

Reduce the number of guesses required by lattice attacks for factoring with partially known bits....

Extract useful information from incorrect guesses in factoring attacks based on partially known bits....

Efficiently solve for, or characterize all relevant roots of, $x^2-1$ modulo a composite integer $N$ in the setting of the slides....

Improve the running time of Coppersmith-type methods for finding small modular or integer roots....

Understand and control how the shape of a polynomial affects Coppersmith-type small-root algorithms....

Give conditions or constructions that ensure algebraic independence in multivariate elimination for small-root attacks....

Find an optimal collection of polynomials for multivariate lattice-based small-root attacks....

Determine whether the lattice dimension in the stated small-root constructions can be reduced....