In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $1<c<\frac{427}{400}$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$, such that $p_1=x^2 + y^2 +1$. For this purpose we establish a new Bombieri -- Vinogradov type result for exponential sums over primes.
Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x + b^y = c^z$.
Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, by analyzing the gap rule for solutions of the ternary purely exponential diophantine equation $a^x+b^y=c^z$, we prove that if $\max\{a,b,c\}\geq 10^{62}$, then the equation has at most two positive integer solutions $(x,y,z)$.
Let k\geq 2 and consider the Diophantine inequality |x_1^k-\alp_2 x_2^k-\alp_3 x_3^k| <\tet. Our goal is to find non-trivial solutions in the variables x_i, 1\leq i\leq 3, all of size about P, assuming that \tet is sufficiently large. We study this problem on average over \alp_3 and generalize previous work of Bourgain on quadratic ternary diagonal forms to general degree k.
Let $c$ be fixed with $1<c<35/34$. In this paper we prove that for every sufficiently large real number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} is solvable in primes $p_1,\,p_2,\,p_3$ near to squares.
Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, combining the Gel'fond-Baker method with an elementary approach, we prove that if $\max\{a,b,c\}>5\times 10^{27}$, then the equation $a^x+b^y=c^z$ has at most three positive integer solutions $(x,y,z)$.
Let $\ell, m, r$ be fixed positive integers such that $2\nmid \ell$, $3\nmid \ell m$, $\ell>r$ and $3\mid r$. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if $\min\{r\ell m^2-1,(\ell-r)\ell m^2+1\}>30$, then the equation $(r\ell m^2-1)^x+((\ell -r)\ell m^2+1)^y=(\ell m)^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$.
In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. Firstly, by assuming some standard modularity conjecture we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Secondly, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\Q(\sqrt{-d})$ where $d=1,7,19,43,67$ whenever $p$ is bigger than this bound. During the course of the proof we prove various results about …
Let $1 < k < 33 / 29$. We prove that if $λ_1$, $λ_2$ and $λ_3$ are non-zero real numbers, not all of the same sign and that $λ_1 / λ_2$ is irrational and $\varpi$ is any real number, then for any $\eps > 0$ the inequality $ \bigl|λ_1 p_1 + λ_2 p_2^2 + λ_3 p_3^k + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-(33 - 29 k) / (72 k) + \eps} $ has infinitely many solutions in prime variables $p_1$, ..., $p_k$.
In this note we will analyze a diophantine equation raised by Michael Bennett in [1] that is pivotal in establishing that powers of five has few digits in its ternary expansion. We will show that the Diophantine equation $3^{a}+3^{b}+2=n^5$, where $(n,3)=1$ and $a>b>0$ is insoluble for pairs of positive integers $(a,b)$ where they are both even or one is even and the other is odd. In the case where both $(a,b)$ are odd, there is one known solution $2^5=3^3+3^1+2.$ We will show that there are no other solutions to the diophantine equation for $n^{5}<32\left(1+3(10^6)\right)^5$.
Let $S$ denote a set of primes and let $a_1,\ldots,a_m$ be positive distinct integers. We call the $m$-tuple $(a_1,\ldots,a_m)$ an $S$-Diophantine tuple if $a_ia_j+1=s_{i,j}$ are $S$-integers for all $i\not=j$. In this paper, we show that no $S$-Diophantine quadruple (i.e~$m=4$) exists if $S=\{2,q\}$ with $q\equiv 3\; (\bmod\, 4)$ or $q<10^9$. For two arbitrary primes $p,q<10^5$ we gain the same result.
Simultaneous Diophantine approximation is concerned with the approximation of a point $\mathbf x\in\mathbb R^d$ by points $\mathbf r\in\mathbb Q^d$, with a view towards jointly minimizing the quantities $\|\mathbf x - \mathbf r\|$ and $H(\mathbf r)$. Here $H(\mathbf r)$ is the so-called "standard height" of the rational point $\mathbf r$. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examp…
Let $\mathfrak{p}=(\mathfrak{p}_1,...,\mathfrak{p}_r)$ be a system of $r$ polynomials with integer coefficients of degree $d$ in $n$ variables $\mathbf{x}=(x_1,...,x_n)$. For a given $r$-tuple of integers, say $\mathbf{s}$, a general local to global type statement is shown via classical Hardy-Littlewood type methods which provides sufficient conditions for the solubility of $\mathfrak{p}(\mathbf{x})=\mathbf{s}$ under the condition that each of the $x_i$'s is prime.
An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the $p$-adic integers for every prime $p$. It is called complete if it is of the form $Q({\mathbf x} + {\mathbf v})$, where $Q$ is an integral quadratic form in the variables ${\mathbf x} = (x_1, \ldots, x_n)$ and ${\mathbf v}$ is a vector in ${\mathbb Q}^n$. Its conductor is defined to be the smallest positive integer $c$ such that $c{\mathbf v} \in {\mathbb Z}^n$. We prove that for a fixed positive integer $c$, there are only finitely many equ…
In this article, we establish an analogue of the dimension growth conjecture, which is regarding the density of rational points on projective varieties, for compact submanifolds of $\mathbb{R}^n$ with non-vanishing curvature. We also establish the convergence theory for the set of simultaneously $ψ$-approximable points lying on a generic hypersurface, thereby settling the generalized Baker-Schmidt problem in the simultaneous setting for generic hypersurfaces. These results are obtained as consequences of an optimal upper bound for the density of rational points near manifolds of the form $\{ (…
Fix $d\in\mathbb N$, and let $S\subseteq\mathbb R^d$ be either a real-analytic manifold or the limit set of an iterated function system (for example, $S$ could be the Cantor set or the von Koch snowflake). An $extrinsic$ Diophantine approximation to a point $\mathbf x\in S$ is a rational point $\mathbf p/q$ close to $\mathbf x$ which lies $outside$ of $S$. These approximations correspond to a question asked by K. Mahler ('84) regarding the Cantor set. Our main result is an extrinsic analogue of Dirichlet's theorem. Specifically, we prove that if $S$ does not contain a line segment, then for ev…
We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exists only finitely many $l$ such that $l!$ is represented {by} $N_A(x)$, where $N_A$ is a norm form constructed from the field norm of a field extension $K/\mathbf Q$. We also deal with the equation $N_A(x)=l!_S$, where $l!_S$ is the Bhargava factorial. In this paper, we also show that the Oesterlé-Masser conjecture implies that for any infinite subset $S$ of $\mathbf Z$ and for any polynomial …
Let $φ_1,\ldots ,φ_r\in \mathbb Z[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text{deg}(φ_j)=k_j$ $(1\le j\le r)$, where $1\le k_1<k_2<\ldots <k_r=k$. Subject to the condition $k_1+\ldots +k_r\ge \tfrac{1}{2}k(k-1)+2$, we show that there is a paucity of non-diagonal solutions to the Diophantine system $φ_j(\mathbf x)=φ_j(\mathbf y)$ $(1\le j\le r)$.
We give upper and lower bounds for various Diophantine exponents associated with the standard linear actions of $\mathrm{SL}_2 ( \mathcal{O}_K )$ on the punctured complex plane $\mathbb{C}^2 \setminus \{ \mathbf{0} \}$, where $K$ is a number field whose ring of integers $\mathcal{O}_K$ is discrete and within a unit distance of any complex number. The results are similar to those of Laurent and Nogueira for $\mathrm{SL}_2 ( \mathbb{Z} )$ action on $\mathbb{R}^2 \setminus \{ \mathbf{0} \}$ albeit for us, uniformly nice bounds are obtained only outside of a set of null measure.
Fix an integer $n\ge 2$. To each non-zero point $\mathbf{u}$ in $\mathbb{R}^n$, one attaches several numbers called exponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point $\mathbf{u}$. To avoid trivialities, one restricts to points $\mathbf{u}$ whose coordinates are linearly independent over $\mathbb{Q}$. The resulting set of values is called the spectrum of these exponents. We show that, in …
We solve a family of quadratic Diophantine equations associated to a simple kind of games. We show that the ternary case, in many ways, is the most interesting and the least arbitrary member of the family.
Let $K= \mathbf{Q}(\sqrt{d})$ be a quadratic field and $\mathcal{O}_{K}$ be its ring of integers. We study the solvability of the Diophantine equation $r + s + t = rst = 2$ in $\mathcal{O}_{K}$. We prove that except for $d= -7, -1, 17$ and $101$ this system is not solvable in the ring of integers of other quadratic fields.
CUNY Graduate Center（纽约市立大学研究生中心）；Oklahoma State University（俄克拉荷马州立大学）；Institute for Advanced Study & Princeton University（普林斯顿高等研究院与普林斯顿大学）；Seoul National University（首尔国立大学）
Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and let $V_F^*$ denote the singular locus of the affine variety $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1, \ldots, x_n) = 0$ provided $F$ satisfies suitable local conditions and $n - \dim V_F^* \geq 2^8 3^4 5^2 d^3 (2d-1)^2 4^{d}$. Our result improves on what was known previously due to Cook and Magyar (B. Cook and A. Magyar, `Diophantine equations in the primes'. Invent. Math. 198 (2…
For any integer $m\ge0$, we recall that triangular numbers are those $\mathbf{T}(m)=\frac{m(m+1)}{2}$. A conjecture of Sun Zhi-Wei states that an integer $2^n\pm n$ with any $n>2$ can not be a triangular number. The motivation of this work is to confirm this conjecture.
We study discrete versions of fractional integral operators along curves and surfaces. $l^p \to l^q$ estimates are obtained from upper bounds of the number of solutions of associated Diophantine systems. In particular, this relates the discrete fractional integral along the curve $γ(m) = (m,m^2,...,m^k)$ to Vinogradov's mean value theorem. Sharp $l^p \to l^q$ estimates of the discrete fractional integral along the hyperbolic paraboloid in $\mathbf{Z}^3$ are also obtained except for endpoints.
Let $α$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field ${\mathbf Q}(α)$. For any unit $ε$ of $K$ such that ${\mathbf Q}(αε)=K$, we consider the irreducible polynomial $f_ε(X)\in{\mathbf Z}[X]$ such that $f_ε(αε)=0$. Let $F_ε(X,Y)\ = Y^df_ε(X/Y)\in{\mathbf Z}[X,Y]$ be the associated binary form. For each positive integer $m$, we exhibit an effectively computable bound for the solutions $(x,y,ε)$ of the diophantine equation $|F_ε(x,y)|\leq m$.
Let $u_k$ be a Lucas sequence. A standard technique for determining the perfect powers in the sequence $u_k$ combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation $u_k=x^n$ can be translated into a ternary equation of the form $a y^2=b x^{2n}+c$ (with $a$, $b$, $c \in \mathbb{Z}$) for which Frey curves are available. In this paper we consider shifted powers in Lucas sequences, and consequently equations of the form $u_k=x^n+c$ which do not typically correspond to ternary equa…
A conjecture of Coleman implies that only finitely many quaternion algebras over the rational numbers can be the endomorphism $\mathbf{Q}$-algebras of abelian surfaces over the complex numbers which can be defined over $\mathbf{Q}$. One may think of this as a higher-dimensional version of the Gauss Class Number problem. Before now, no one has ruled out quaternion algebras over $\mathbf{Q}$ not already ruled out by the Albert classification. We rule out infinitely many such quaternion algebras by showing that for infinitely many $D$, the Atkin-Lehner quotient Shimura curve $X^D/w_D$ has no $\ma…
For a decreasing real valued function $ψ$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $ψ$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m < ψ(T)\quad\text{and}\quad\|\mathbf{q}\|^n < T$$ has a solution $\mathbf{p}\in\mathbb{Z}^m$, $\mathbf{q}\in\mathbb{Z}^n$ for all sufficiently large $T$, where $\|\cdot\|$ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the $ψ$-Dirichlet non-improvable set. In this paper, we prove a similar criter…
