S. I. Dimitrov
Let $\varepsilon>0$ be a small constant. In the present paper we prove that whenever $η$ is real and constants $λ_i$ satisfy some necessary conditions, then there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p_3+η|<\varepsilon \end{equation*} and such that $p_3=x^2 + y^2 +1$.
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Diophantine approximation on lines with prime constraints
研究了R^2中具有素分子和素分母的直线上的丢番图逼近问题。
Diophantine approximation on lines in $\mathbb{C}^2$ with Gaussian prime constraints
本文研究了$\mathbb{C}^2$中分子和子均为高斯素数的直线上的丢番图逼近问题。
On Proving of Diophantine Inequalities with Prime Numbers by Evaluations of the Difference between Consecutive Primes
作为工作假设的评价之间的差异素数$p_{n+1} - p_n = O(\sqrt{p_n})$，我们代表了勒让德和Oppermann的代数的详细证明。
A note on Diophantine approximation with Gaussian primes
研究了$pθ$模1的分布，其中$θ$是不包含在$\mathbb{Q}(i)$中的复数，且$p$在高斯素数上运行.
Diophantine Inequalities as a Problem of Difference between Consecutive Primes
在本文中，我们已经开发了一种方法来解决\textit{diophantine inequalities}使用它们与\textit{difference between consecutive primes}的关系。使用这种方法，我们已经能够证明一些定理，包括英厄姆的指数定理以及一些新的结果。丢番图不等式及其与Cramer和Andrica不等式的关系也被讨论。
Given a finite set of primes $S$ and a $m$-tuple $(a_1,\dots,a_m)$ of positive, distinct integers we call the $m$-tuple $S$-Diophantine, if for each $1\leq i < j\leq m$ the quantity $a_ia_j+1$ has prime divisors coming only from the set $S$. For a given set $S$ we give a practical algorithm to find all $S$-Diophantine quadruples, provided that $|S|=3$.
Diophantine Inequalities with Primes, Auxiliary Inequalities, Evaluations of the Difference between Consecutive Primes
利用辅助不等式和连续素数差的估计，给出了一种证明含素数的丢番图不等式的方法.本文研究了区间$((n - 1)^k, n^k)$中素数的Legendre - Ingham问题以及当$k$为实数时区间$(\frac{k-1}{k}n, \frac{k}{k-1}n)$中素数的Legendre - Ingham问题.一些新的结果，包括一个替代的证明英厄姆定理的有效计算常数，也英厄姆定理与两个素数的证明。
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.
The inequalities concern the sum of s powers of primes with non-integer exponent c>1. Here s =2,3,4,or 5. The equations are similar, taking integer part before summing; here s = 3 or 5. New ranges of c are found in all cases for which many solutions in primes exist.
Diophantine approximation on lines in \mathbb{C}^2 with Gaussian prime constraints - enhanced version
我们研究了分子和分母限制为高斯素数的$\mathbb{C}^2$中直线上的丢番图逼近问题。为此，我们开发了类似物的知名结果的小分数部分的$pγ$，$p$运行的素数和$γ$是一个固定的不合理的，高斯素数。
Let $[x]$ denote the integral part of the real number $x$, and $N$ be a sufficiently large integer. In this paper, it is proved that, for $1<c<\frac{4109054}{1999527}, c\not=2$, the Diophantine equation $N=[p_1^c]+[p_2^c]+[p_3^c]+[p_4^c]+[p_5^c]$ is solvable in prime variables $p_1,p_2,p_3,p_4,p_5$.
General Solution of the Diophantine equation involving Mersenne Prime
该研究针对含梅森素数的指数丢番图方程$M_p^{x} + (M_q + 1)^{y}= (lz)^2$展开求解，给出了方程的解的相关示例与无解的情形，丰富了梅森素数相关丢番图方程的研究成果。
On Two Diophantine Inequalities Over Primes
设$1<c<37/18,\,c\neq2$和$N$是足够大的实数。本文证明了对几乎所有的$R\in(N,2N],$，丢番图不等式$|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$在素数$p_1,\,p_2,\,p_3.$中可解。此外，我们还研究了六个素数的问题，证明了对充分大的实数$N$，丢番图不等式$|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N|<\log^{-1}N$在素数$p_1,\,p_2,\,p_3,\,p_4,\,p_5,\,p_6$中可解。
Diophantine Approximation with Products of Two Primes
证明了对任意无理数$α$和任意$τ<8/23$，都有无穷多个$n$是两个素数的乘积，其中$$\|nα\|\leq n^{-τ}.$$.我们还证明了对任意充分大的$b$，都存在三位回文在$b$基上，恰好有两个素数因子.
We show that whenever $δ>0$, $η$ is real and constants $λ_i$ satisfy some necessary conditions, there are infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality $|λ_1p_1 + λ_2p_2 + λ_3p_3+η|<(\max p_j)^{-1/12+δ}$ and such that, for each $i\in\{1,2,3\}$, $p_i+2$ has at most $28$ prime factors.
Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<\log^{-1}N \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4$, which improves the result of Mu.
On a diophantine inequality with prime numbers of a special type
考虑丢番图不等式\[ \left| p_1^{c} + p_2^{c} + p_3^c- N \right| < (\log N)^{-E} , \]，其中$1 < c < \frac{15}{14}$，$N$是一个充分大的实数，$E>0$是一个任意大的常数.我们证明了上述不等式在素数$p_1$，$p_2$，$p_3$中有解，使得每个数$p_1 + 2, p_2 + 2, p_3 + 2$至多有$\left[ \frac{369}{180 - 168 c} \right]$素因子，以重数计数.
A quaternary diophantine inequality by prime numbers of a special type
令$1<c<832/825$。对于大实数$N>0$和小常数$\vartheta>0$，不等式\begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\vartheta \end{equation*}在素数$p_1,\,p_2,\,p_3,\,p_4$中有解，使得对于每个$i\in\{1,2,3,4\}$，$p_i+2$至多有$32$素因子。
Prime Solutions of Diagonal Diophantine Systems
该研究通过建立对角丢番图方程组素数解的渐近公式，结合辛格罗多夫定理和霍利工作，给出了沃林-戈尔德巴奇问题的条件结果。
A Diophantine problem with a prime and three squares of primes
证明了：若$λ_1$，$λ_2$，$λ_3$和$λ_4$是非零实数，且符号不相同，$λ_1 / λ_2$是无理数，$\varpi$是任意实数，则对任意$\eps > 0$，不等式$ \bigl|λ_1 p_1 + λ_2 p_2^2 + λ_3 p_3^2 + λ_4 p_4^2 + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-1 / 18 + \eps} $在素变量$p_1$，.中有无穷多个解，$p_4$
A Diophantine approximation problem with two primes and one $k$-th power of a prime
本文改进了两个素数和一个素数k次幂的丢番图逼近问题，将k的范围扩展到1 < k ≤ 3，并给出了更紧的逼近界。
Let $S$ be a set of primes. We call an $m$-tuple $(a_1,\ldots,a_m)$ of distinct, positive integers $S$-Diophantine, if for all $i\neq j$ the integers $s_{i,j}:=a_ia_j+1$ have only prime divisors coming from the set $S$, i.e. if all $s_{i,j}$ are $S$-units. In this paper, we show that no $S$-Diophantine quadruple (i.e.~$m=4$) exists if $S=\{3,q\}$. Furthermore we show that for all pairs of primes $(p,q)$ with $p<q$ and $p\equiv 3\mod 4$ no $\{p,q\}$-Diophantine quadruples exist, provided that $(p,q)$ is not a Wieferich prime pair.
Almost prime solutions to diophantine systems of high rank
设$\F$是一族$k\geq 2$积分形式，$\LL=(l_1,\ldots,l_m)$是一族关于$n$变量$\x=(x_1,...,x_n)$的两两线性无关线性形式.本文研究了丢番图系统$\F(\x)=\vv$在$l_i(\x)$对每个$1\leq i\leq m$有有界素因子的限制下的解的个数$\x\in[1,N]^n$.我们表明，系统$\F$有预期数量的“几乎素”的解决方案，在类似的条件下建立的整数解的存在性由伯奇。
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…
Let $1<c<\frac{26088036}{12301745},c\not=2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R\in (N,2N]$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c-R\big|<\log^{-1}N \end{equation*} is solvable in primes $p_1,p_2,p_3$. Moreover, we also prove that the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N\big|<\log^{-1}N \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4,p_5,p_6$, which improves the previous result $1<c<\frac{37}{18},c\neq2$.
On a Diophantine problem with one prime, two squares of primes and $s$ powers of two
本文改进了W.P. Li和Wang关于$ λ_1p_1 + λ_2p_2^{2} + λ_3p_3^{2} + μ_1 2^{m_1} +...+ μ_s 2^{m_s}, $形式的值的一个结果，其中$p_1,p_2,p_3$是素数，$m_1,..., m_s$是正整数，$λ_1,λ_2,λ_{3}$是非零实数，不都是同号的，$λ_2 / λ_3$是无理数，而$λ_i/μ_i \in \Q$，对于$i\in\{1,2,3\}$ .
Diophantine equations in the primes
令$\mathfrak{p}=(\mathfrak{p}_1,...,\mathfrak{p}_r)$是$r$多项式的系统，其具有$d$在$n$变量$\mathbf{x}=(x_1,...,x_n)$中的整数次数系数。对于给定的整数$r$ -元组，比如$\mathbf{s}$，通过经典的Hardy-Littlewood型方法给出了一个一般的局部到全局类型语句，该语句在每个$x_i$都是素数的条件下为$\mathfrak{p}(\mathbf{x})=\mathbf{s}$的可解性提供了充分条件.
On a Diophantine inequality involving a prime and an almost-prime
证明了当$r=3,$、$τ=\frac1{118}$和$λ_0$为任意实数时，$$ |λ_0+λ_1p+λ_2P_r|<p^{-τ}, $$有无穷多个解，且$λ_1,λ_2\in\BR$、$λ_2\neq0$和$0>\frac{λ_1}{λ_2}$不在$\mathbb{Q}$中.这改进了Harman的一个结果。此外，我们还证明了对于某个正整数$n$，可以要求素数$p$的形式为$\floor{n^c}$，即$p$是Piatetski-Shapiro素数，$r=13$和$τ=ρ(c),$是由$\left(1, 1+\frac1{149}\right].$中支持的$c$显式确定的常数
Let $[\,\cdot\,]$ be the floor function. In this paper we show that whenever $η$ is real, the constants $λ_i$ satisfy some necessary conditions, then for any fixed $1<c<38/37$ there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p_3+η|<(\max p_j)^{{\frac{37c-38}{26c}}}(\log\max p_j)^{10} \end{equation*} and such that $p_i=[n_i^c]$, $i=1,\,2,\,3$.
On a Diophantine problem with two primes and s powers of two
我们改进了Parsell最近的结果的值的形式$λ_1p_1 + λ_2p_2 + μ_1 2^{m_1} + ...m + μ_s 2^{m_s}, $其中$p_1,p_2$是素数，$m_1,...c, m_s$是正整数，$λ_1 / λ_2$是负的和无理数和$λ_1 / μ_1$，$λ_2/μ_2 \in \Q$。
A Diophantine problem with prime variables
本文研究了$λ_1 p_1 + λ_2 p_2 + λ_3 p_3^k$型值的分布，其中$λ_1$，$λ_2$和$λ_3$都是非零实数，符号不同，$λ_1 / λ_2$是无理数，$p_1$，$p_2$和$p_3$是素数.我们证明，当$1 < k < 4 / 3$，这些值近似而密切任何规定的实数。
Intrinsic Diophantine approximation by rationals of height with a bounded number of distinct prime factors
本文研究了由具有有限不同素因子数的有理数逼近的自相似集的Hausdorff维数，修正了前一版本中引理3.8的证明错误。
Random Diophantine Equations in the Primes II
研究素数中高斯丢番图方程的局部-全局原理，采用Browning等人的方法扩展前人工作。
Diophantine approximation with prime denominator in quadratic number fields under GRH
本文在广义黎曼猜想对Hecke $L$-函数成立的条件下，将Matomäki关于实数无理数素数分母丢番图逼近的结果推广至所有二次数域。
Diophantine Approximation with Piatetski-Shapiro Primes
本文证明了关于Piatetski-Shapiro素数的丢番图逼近结果，即在特定参数条件下存在无穷多个形如$p=[n^c]$的素数满足给定的逼近不等式。
Elliptic curves of twin-primes over Gauss field and Diophantine Equations
设$p, q$与$q-p=2$是孪生素数.考虑椭圆曲线E=E_σ：y^2 = x（x+σp）（x+σq）.（σ= \pm 1）。当σ=+1或$-1时，E=E_σ也记为E_+或E_-.这里确定了高斯域K=Q（\sqrt-1）（和有理域Q）上椭圆曲线E的Mordell-Weil群和秩，并给出了有关的丢番图方程和联立Pellian方程解的结果.上一篇文章[1]研究了Q上椭圆曲线E的算术结构，确定了Selmer群，得到了关于Mordell-Weil群、秩、Shafarevich-Tate群和扭子群的结果.
Random Diophantine Equations in the Primes
研究素数解的存在性，提出素数哈塞原理，证明当s≥3k+2时几乎所有此类方程有解，并给出反例显示哈塞-米诺夫斯基定理的素数解版本失效。
Solving diophantine equations x^4 + y^4 = q z^p
对于$q$的某些素数和每个大于13的素数$p$，给出了求解$x^4 + y^4 = q z^p$型广义Fermat方程的一种方法.我们通过证明$q= 73, 89$和113没有解决方案来说明该方法。
Diophantine approximation with primes from short intervals
本文研究了短区间中素数在丢番图逼近中的渐进行为，证明了当素数p满足||pα||<δ时，其数量渐近等于2δY/log X。
On a ternary Diophantine problem with mixed powers of primes
令$1 < k < 33 / 29$。本文证明了：如果$λ_1$，$λ_2$和$λ_3$都是非零实数，但符号不同，且$λ_1 / λ_2$是无理数，$\varpi$是任意实数，则对任意$\eps > 0$，不等式$ \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} $在素变元$p_1$，.中有无穷多个解，$p_k$。
In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors on Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)\in\mathbb Z[x]$, $a$ a positive integer and $b$ an arbitrary positive integer.
On Iwasawa theory of abelian varieties over $\mathbb{Z}_p^2$-extension with applications to Diophantine stability and integally Diophantine extensions
本文研究Abel曲线上Iwasawa理论，探讨丢番图稳定性及整数丢番图扩张，并改进Mazur猜想相关结果，扩展至具有良好超奇异减的椭圆曲线。
On a Diophantine Inequality with Primes Yielding Square-Free Sums with Given Numbers
本文证明对任意无理数α和实数β，存在无穷多个素数p满足αp+β的分数部分小于p^{-1/10}，且p与给定互异正整数之和均为无平方数。
Diophantine approximation with one prime, two squares of primes and one $k$-th power of a prime
本研究针对特定范围的$k$与满足条件的非零实系数，证明含一个素数、两个素数平方及一个素数$k$次幂的线性型逼近实数的不等式有无穷多素数解，推进了丢番图逼近领域的相关研究。
A ternary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$
本文证明对任意 $1<c<427/400$，三元 Piatetski-Shapiro 不等式在素数中有解，且其中一个素数可表示为 $x^2+y^2+1$，为此建立了新的 Bombieri--Vinogradov 型指数和结果。
Arithmetic statistics and diophantine stability for elliptic curves
本文研究有理数域椭圆曲线在素数幂循环扩张中的Mordell-Weil群与Tate-Shafarevich群稳定性，提出“Sha”-稳定性概念，并利用算术统计与岩泽理论证明非CM椭圆曲线在几乎所有素数处稳定。
On a binary Diophantine inequality involving primes of a special type
本文研究了涉及特殊类型素数的丢番图不等式可解性问题，证明了二元与四元情形下，满足对应素因子个数上界条件的素数解的存在性，给出了素因子数的具体上界公式。
S-Diophantine quadruples with $\mathbf{S=\{2,q\}}$
令$S$表示素数集合，令$a_1,\ldots,a_m$是正的不同整数。如果对于所有$i\not=j$，$a_ia_j+1=s_{i,j}$都是$S$ -整数，我们称$m$ -元组$(a_1,\ldots,a_m)$为$S$ -丢番图元组.本文证明了，如果$S=\{2,q\}$与$q\equiv 3\; (\bmod\, 4)$或$q<10^9$，则不存在$S$ -丢番图四元组（即~ $m=4$）.对于两个任意素数$p,q<10^5$，我们得到了同样的结果.
Diophantine approximation with mixed powers of Piatetski-Shapiro primes
本文研究了混合幂Piatetski-Shapiro素数的丢番图逼近问题，证明在特定条件下存在无限多个素数三元组满足特定不等式。
In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $pα$ when $α$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to the nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered t…
The exceptional set for Diophantine inequality with mixed powers of primes
研究混合素数幂的丢番图不等式超越集的上界，采用非退化序列和正数δ，确定素数解不存在的上界。
A quinary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$
本文证明了在 $1<c<\frac{5363}{3900}$ 时，存在素数解满足五元丢番图不等式，且其中一素数具有 $p=x^2+y^2+1$ 的形式。
Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, the maximal abelian extension of $\mathbb{Q}$. We show that there exists an algorithm to decide whether a system of equations and inequations, with integer coefficients, has a solution in $\mathbb{Z}^{ab}$ modulo every rational prime.
Given $n\in N$ and $x,γ\in R$, let \begin{equation*} ||γ-nx||^\prime=\min\{|γ-nx+m|:m\in Z, \gcd (n,m)=1\}, \end{equation*} %where $(n,m)$ is the largest common divisor of $n$ and $m$. Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number $α$ and almost every $γ\in R$, \begin{equation*} \liminf_{n\to \infty}n||γ-nα||^{\prime}=0 \end{equation*} and that there exists $C>0$, such that for all $α\in R\backslash Q$ and $γ\in [0,1)$ , \begin{equation*} \liminf_{n\to \infty}n||γ-nα||^{\prime} < C. \end{equation*} We prove the first conjecture and …
Diophantine approximation by primes and Landau--Siegel zeros
本文在假设Siegel零点存在或足够强的条件下，证明了有限型无理数$\alpha$与实数$\beta$的线性组合$\alpha p+\beta$可被素数$p$逼近到$p^{-1/3+\varepsilon}$或$p^{-1/3-1/20}$的精度，给出了素数丢番图逼近的新结果。
Small prime solutions to cubic Diophantine equations
设a1，...，a9是非零整数，n是任何整数。假设a1 +…+ a9 = n（mod 2）和（ai，aj）= 1，其中1 <= i < j <= 9。我们将证明：（i）如果不是所有aj的符号都相同，则三次对角方程a1p1^3 +. + a9p9^3 = n有素解满足pj << n^{1/3} + max{ |aj| }^{20+ e};（ii）若所有aj均为正且n >> max{ |aj| }^{61+e}，则三次对角方程a1p1^3 +. + a9p9^3 = n可溶于素数pj。这个结果是由S.K.K. Choi和J. Liu。为了证明这些结果，我们将使用Hardy-Littlewood圆方法，我们将概述。最后，我们将对这个特殊问题的一些可能的推广做一个说明。
Intersective polynomials and Diophantine approximation, II
应用Schmidt格方法，证明了在满足一定局部条件下，单素变量多项式系统模1的同时丢番图逼近的结果.
Erdős considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes $p$ for which there is no $\mathbb{Q}_p$-point on a random variety are Poisson distributed.
Exponents of Diophantine approximation
本文主要综述了丢番图逼近的最新结果，着重讨论了各种丢番图逼近概念的指数度量的性质。
Let $p$ be a prime integer, $\mathbb{Z}_p$ the finite field of order $p$ and $\mathbb{Z}^{*}_{p}$ is its multiplicative cyclic group. We consider the Diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq \frac{p - 1}{2}$. Our main aim in this paper is to give optimal conditions or relationships between the exponent $n$ and the prime $p$ to determine the existence of nontrivial solutions of the diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq p -1 $, in finite fields $\mathbb{Z}_p$.
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