Mathematics Problem Archive

If $a$ and $b$ are two positive integers with $a<b$ and $R(a)=R(b)$ then there is a prime $p$ with $a< p< b$....

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 $(\varepsilon_n)_{n\ge 0}$ be a sequence of elements in $\{0,1\}$. Assume that the real number $$ \sum_{n\ge 0}\varepsilon_n 3^{-n} $$ is irration...

Let $k$ 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...

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 $\mathbb{Q}$....

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

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

Let $\tau\in\mathbb{C}$ have positive imaginary part. Assume that $\tau$ is not quadratic. Set $q=e^{2i\pi\tau}$. Then at least $4$ of the $5$ numbers...

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 each integer $n\ge 1$ there exists a positive constant $c(n)$ such that, for any algebraic subvariety $V$ of $\mathbb{G}_m^n$ which is defined ove...

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

There exists a positive constant $c$, depending only on $A$ and $\mathcal{L}$, such that for any $P\in A(\overline{\mathbb{Q}})$ which has infinite or...

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 $\underline{\theta}=(\theta_1,\ldots,\theta_m)$ 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...

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

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 $m$, $n$, $k$ 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 $n \ge 2$. Denote by $\mathrm{ZH}_n$ the set of real numbers $\xi$ with the following property: there exists $c_1(\xi) >0$ and $c_2(\xi)>0$ such t...