S. I. Dimitrov
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[αn]$, $[αn]+1$, where $α>1$ is irrational number with bounded partial quotient or irrational algebraic number.
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In this short paper we shall prove that there exist infinitely many consecutive square-free numbers of the form $[αp]$, $[αp]+1$, where $p$ is prime and $α>0$ is irrational algebraic number. We also establish an asymptotic formula for the number of such square-free pairs when $p$ does not exceed given sufficiently large positive integer.
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $n^2+1$, $n^2+2$. We also establish an asymptotic formula for the number of such square-free pairs when $n$ does not exceed given sufficiently large positive integer.
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
Perfect powers that are sums of consecutive squares
我们确定所有可以写成最多10个连续平方和的完美幂。
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $x^2+y^2+1$, $x^2+y^2+2$. We also give an asymptotic formula for the number of pairs of positive integers $x, y \leq H$ such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free.
Difference of powers of consecutive primes which are perfect squares
研究了连续素数$p$和$q$（$p > q$），对它们存在一对自然数$(x,y)$使得$p^x-q^y$是完满平方，并作了一些证明.
In the present paper we prove that for any fixed $1<c<7/6$ there exist infinitely many consecutive square-free numbers of the form $[n^c], [n^c]+1$ and we also establish an asymptotic formula in given interval.
Large gaps between consecutive prime numbers containing square-free numbers and perfect powers of prime numbers
我们证明了K. Ford，D. R. Heath-Brown和S. Konyagin关于无平方数的素数回避和素数的完全幂。
On sequences of consecutive squares on elliptic curves
令$C$是由方程$y^2=x^3+Ax+B$在$\mathbb Q$上定义的椭圆曲线，其中$A,B\in\mathbb Q$。一个有理点序列$(x_i,y_i)\in C(\mathbb Q),\,i=1,2,\ldots,$被称为在$C$上形成一个连续的正方形序列，如果$x$ -坐标序列$x_i,i=1,2,\ldots$由连续的正方形组成。我们产生了一个无限族的椭圆曲线$C$与$5$ -项序列的连续平方。此外，该序列由$C(\mathbb Q)$中的五个独立有理点组成。具体地，$C(\mathbb Q)$的秩$r$满足$r\ge 5$。
In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy--Littlewood conjectures containing a large secondary term, which fits the data very well. In this paper, we study consecutive pairs of sums of squares in arithmetic progressions, and develop a similar heuristic model based on the Hardy--Littlewood conjecture for sums of squares, which also explain the biases in the experimental data. In the process, we prove several …
Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers
考虑寻找$M$连续整数平方和的所有整数解的问题，从$a^{2}$开始等于平方整数$s^{2}$，它表明，如果$M\equiv3,5,6,7,8$或$10 (mod\,12)$没有解决方案，并有整数解，如果$M\equiv0,9,24$或$33 (mod\,72)$ ;或$M\equiv1,2$或$16 (mod\,24)$ ;或$M\equiv11 (mod\,12)$。$M$的所有允许值都使用必要条件来表征。如果$M$本身是一个正方形，那么$M\equiv1 (mod\,24)$和$(M-1)/24$都是五边形数，除了前两个。
An infinite collection of quartic polynomials whose products of consecutive values are not perfect squares
利用一个初等恒等式，证明了对无穷多个四次多项式$P(x)\in \mathbb{Z}[X]$，方程$\prod\limits_{k=1}^{n}P(k)=y^2$在$\mathbb{Z}$中有无穷多个解.我们还给出了一个四次多项式的例子，它的第一个连续值的乘积往往是一个完满平方。
Let $C: y^2=ax^4+bx^2+c$, be an elliptic curve defined over $\mathbb Q$. A set of rational points $(x_i,y_i) \in C(\mathbb Q)$, $i=1,2,\cdots,$ is said to be a sequence of consecutive squares if $x_i= (u + i)^2$, $i=1,2,\cdots$, for some $u\in \mathbb Q$. Using ideas of Mestre, we construct infinitely many elliptic curves $C$ with sequences of consecutive squares of length at least $6$. It turns out that these 6 rational points are independent. We then strengthen this result by proving that for a fixed $6$-term sequence of consecutive squares, there are infinitely many elliptic curves $C$ with…
In this note, we prove by using T. Estermann's and S. Dimitrov's arguments with an elementary inequality that there are infinitely many $n$ for which all of the numbers $n^2+1,n^2+2$ and $n^2+3$ are squarefree. We also improve the error term slightly in the case of two consecutive numbers of the same form, so that we are able to prove the following asymptotic formula. \begin{align*} \sum_{n\le X}μ^2(n^2+1)μ^2(n^2+2)μ^2(n^2+3)\sim\dfrac{7}{18}\prod_{p>3}\left(1-\dfrac{3+\left(\frac{-1}{p}\right)+\left(\frac{-2}{p}\right)+\left(\frac{-3}{p}\right)}{p^2}\right)X. \end{align*}
Consecutive square-free values for some polynomials
该研究针对两类多项式形式的连续无平方因子数的分布问题展开分析，为两种情形分别建立了对应的渐近公式，丰富了数论中多项式无平方因子值的相关研究结果。
In this note, we study the problem of existence of sequences of consecutive 1's in the periodic part of the continued fractions expansions of square roots of primes. We prove unconditionally that, for a given $N\gg 1$, there are at least $N\log^{-3/2}N$ prime numbers $p\leq N$ such that the continued fraction expansion of $\sqrt{p}$ contains three consecutive 1's on the beginning of the periodic part. We also present results of our computations related to the considered problem and some related problems, formulate several open questions and conjectures and get some results under the assumption…
Positive density for consecutive runs of sums of two squares
本文研究了两个平方数之和的连续序列在等差数列中的分布，证明对于任意奇数无平方因子模数和简化同余类，存在正密度的此类连续序列，这是 Maynard 素数序列结果的类比。
Arithmetic Polygons and Sums of Consecutive Squares
本文研究了算术多边形与连续平方和的关系，证明了对于奇数N≥3存在至少一个N边算术多边形，并展示了存在无限多个偶数边的算术多边形。
Three consecutive almost squares
给定一个正整数$n$，我们让${\rm sfp}(n)$表示$n$的无平方部分。我们确定所有的正整数$n$，其中$\max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} \leq 150$有关的问题，以寻找积分点的椭圆曲线。我们还证明了存在无穷多个$n$，其中\[ \max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} < n^{1/3}. \]
Square-free values of $\mathbf{n^2+n+1}$
本文通过改进余项推导渐近公式，证明了形如 $n^2+n+1$ 的数中存在无穷多个无平方数。
Nonexistence of Consecutive Powerful Triplets Around Cubes with Prime-Square Factors
本文证明了围绕具有素数平方因子的立方体的连续强力三元组不存在，结合模算术、p-adic赋值、Thue方程和椭圆曲线理论，推进了Erdős-Mollin-Walsh猜想的研究。
Congruent conditions on the number of terms, on the ratio number of terms to first terms and on the difference of first terms for sums of consecutive squared integers equal to squared integers
$M$连续平方整数$\left(a+i\right)^{2}$等于平方整数（对于$a\geq1$，$0\leq i\leq M-1$）的和产生$M$连续相同值的$a$值对$\left(a_{1},a_{2}\right)$的某些线性分组，当这些通过$a_{1}+a_{2}=μM+1$与$μ=\left(η/δ\right)\in\mathbb{\mathbb{Q}}^{+}$链接时。在本文中，合同的条件$M,η,δ$，并在差异$\left(a_{2}-a_{1}\right)$证明这些线性分组举行。发现$η\equiv1\left(mod\,2\right)$和$δ\equiv0,1$或$5\left(mod\,6\right)$，如果$δ\equiv0\left(mod\,6\right)$，则为$M\equiv0\left(mod\,12\right)$，而如果$δ\equiv1$或$5\left(mod\,6\right)$，则为$M\equiv2$或$11\left(mod\,12\right)$，其中$a_{1}$和$a_{2}$具有不同或相同的奇偶性。
Bounded gaps and perfect power gaps in sequences of consecutive primes
本文研究连续素数间隙中是否存在较大间隙或完美幂间隙，并探讨相关数学问题。
General solutions of sums of consecutive cubed integers equal to squared integers
通过将$\left(a+M-1\right)$和$\left(a-1\right)$的三角数的差与和的乘积分解为它们的最大公约数$g$和剩余平方因子$δ^{2}$和$σ^{2}$的乘积，得到$c=gδσ$，从而找到$M$连续立方整数$\left(a+i\right)^{3}$（$a>1$，$0\leq i\leq M-1$）的和等于平方整数$c^{2}$的问题的所有整数解$\left(M,a,c\right)$。此外，$g$必须是整数的几个特定的和一般的情况下产生广义佩尔方程的解决方案允许找到所有整数解决方案$\left(M,a,c\right)$，这些解决方案出现递归的条件。特别是，发现对于$M$的所有奇数值、$a$的所有奇数整数平方值以及$M$的所有偶数值等于整数平方的两倍的情况，总是存在至少一个解。
On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares
本文通过改进Dudek和Johnston的结果，证明了每对连续平方之间存在一个具有最多3个素因子的整数，从而在素数分布理论中取得进展。
On consecutive quadratic non-residues: a conjecture of Issai Schur
Issai Schur曾经问过是否有可能确定一个界限，最好是使用初等方法，这样对于所有素数p大于界限，模p的连续二次非剩余数的最大个数总是小于p的平方根.本文用初等方法证明了13是唯一一个连续二次非剩余数的最大个数小于p的平方根的素数.模p的余数超过p的平方根。
The numbers of representations of totally positive integers as sums of three integer squares in $\mathbf{Q}(\sqrt{3})$ and in $\mathbf{Q}(\sqrt{17})$, are studied by using Shimura lifting map of Hilbert modular forms. We show the following results. In case of $\mathbf{Q}(\sqrt{3})$, a totally positive integer $a+b\sqrt{3}$ is represented as a sum of three integer squares if and only if $b$ is even. In case of $\mathbf{Q}(\sqrt{17})$, a totally positive integer is represented as a sum of three integer squares if and only if it is not in the form $π_{2}^{2e}π_{2}'^{2e'}μ$ with $μ\equiv7\pmod{π_{…
Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials
如果是$M\equiv0,9,24$或$33(mod\,72)$ ;或$M\equiv1,2$或$16(mod\,24)$;或$M\equiv11(mod\,12)$，则从$a^{2}\geq1$开始的$M$连续整数平方和的平方根$s$为整数，如果是$M\equiv3,5,6,7,8$或$10(mod\,12)$，则不能为整数。求出所有$s$整数的解需要求解以$M$为参数的变量为$a$和$s$的丢番图二次方程。如果$M$不是平方整数，则变量为$a$和$s$的丢番图二次方程被变换为广义佩尔方程，其形式取决于$M(mod\,4)$的同余值，并且其解（如果存在）对于给定的$M$值产生$a$和$s$中的所有解。根据这个广义Pell方程是否存在一个或多个基本解，在$a$和$s$中存在一个或多个解的无限分支，这些解可以简单地写成在相关简单Pell方程的基本解处求值的切比雪夫多项式的函数.如果$M$是一个平方整数，它是已知的，$M\equiv1(mod\,24)$和$M=(6n-1)^{2}$的所有整数$n$ ;然后丢番图二次方程的变量$a$和$s$减少到一个简单的差异的整数平方产生一个有限数目的解决方案在$a$和$s$的初始问题。
On moments of gaps between consecutive squarefree numbers
本文研究了连续无平方因子数之间间隙的矩，证明了相关渐近式在 $0 \le γ< 3.75$ 范围内成立，将此前 $0 \le γ< 3.6875$ 的最佳范围予以改进。
Consecutive non-square non-primitive tuples in finite fields
研究有限域中连续非平方非本原元组问题，通过得到非平方\(\ell\)次幂连续元组存在性定理并结合有限计算，证明\(\theta_q<4/15\)保证有三个连续NSNP元素，且\(4/15\)最优。
Additional congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers
在$a$、$M$和$s$中寻找从$a^{2}\geq1$开始的$M$连续整数平方和等于平方整数$s^{2}$的所有整数解的问题，如果是$M\equiv3,5,6,7,8$或$10\left(mod\,12\right)$则没有解，如果是$M\equiv0,9,24% or %33\left(mod\,72\right)$ ;或$M\equiv1,2$或$16\left(mod\,24\right)$ ;或$M\equiv11\left(mod\,12\right)$则有整数解。在本文中，额外的合同条件证明了对允许值的$M$，产生的解决方案，通过使用Beeckmans的八个必要条件，进一步细化的可能值的$M$的$M$连续整数平方和等于整数平方。
Metrical properties of weighted products of consecutive Lüroth digits
该研究针对连续吕罗特数字加权乘积构成的例外集，在给定条件下证明了勒贝格测度零一律，并分情形计算了该集合的豪斯多夫维数。
Reducing Every Set of 36 Consecutive Integers to Zero by Differences of Squares
该研究解决了Hickerson与Kleber提出的问题，给出将任意36个连续整数通过平方差化为零的明确方法，结合已有结果得出整数集可对所有整数n化为零的长度条件为12整除且不小于24。
Triples and quadruples of consecutive squares or non-squares in a finite field
本文研究有限域中连续平方数或非平方数的三元组与四元组计数，利用 Jacobsthal 和求值给出显式公式，推广了 Monzingo 的结果，并处理了 5 的幂阶情形。
Pairs of k-free Numbers, consecutive square-full Numbers
我们考虑了直到$x$的无$k$整数对的数目的渐近公式的误差项。我们的误差项改进了Heath-Brown，Brandes和Dietmann/Marmon的结果。然后，我们将我们的结果推广到$r$ -元组的$k$ -免费的号码和改进以前的结果曾。此外，我们建立了一个误差项连续平方满整数。最后，我们将证明，对于所有的$θ<3$和几乎所有的$D$，与Pell方程$x^2-Dy^2=1$相关联的基本解$ε_D$满足$ε_D> D^θ$。这改进/恢复了Fouvry和Jouve之前的结果。我们工作的主要工具是近似行列式方法。
The Frobenius number corresponding to the squares of three consecutive Fibonacci numbers: comparison of three algorithmic processes
本文针对三个连续斐波那契数平方生成的数值半群，计算其Frobenius数，并比较Ramírez Alfonsín-Rødseth、Rosales-García-Sánchez和Tripathi三种算法过程的效果。
On sums of powers of consecutive squares over finite fields, and sums of distinct values of polynomials
本文研究有限域中连续元素为平方或非平方时的幂和，并将其用于计算四次多项式 X^4+aX^2+b 的像集元素之和，解决了两个已有猜想。
The Arithmetic Geometry of Square-Sided Heron Triangles
本文研究了具有三个完全平方边的海伦三角形的等价条件，并通过代数曲线的有理点来确定其有限性，同时给出了具有两个完全平方边的海伦三角形的完整刻画及参数解族，证明了存在无限多个原始海伦三角形。
Consecutive runs of sums of two squares
本文研究两个平方数之和序列在算术级数中的连续分布，证明了在可容许同余类条件下，任意长度三段的同余模式以及两段任意长度的同余模式均可出现无穷多次。
Sums of primes and squares of primes in short intervals
令$\mathbf H_2$表示偶数集合$n \not\equiv 1 \pmod 3$。我们证明了当$H \ge X^{0.33}$，几乎所有的整数$n \in \mathbf H_2$，$X < n \le X + H$都可以表示为一个素数与一个素数的平方之和.我们也证明了一个类似的结果，三个平方和的素数。
A Dynamical Analogue of a Question of Fermat
本文研究二次多项式迭代轨道中的连续平方数问题，构造了含三个连续平方数的单参数族，并证明存在x^2+c形式的多项式其轨道含四个连续平方数，同时在标准猜想下给出周期点轨道仅含有理平方数的充要条件。
On integral bases and monogenity of pure octic number fields with non-square free parameters
本文研究了非平方自由参数下纯八次数域的单本性，通过计算其整数环的整基，刻画了整基等于 $\mathbf Z[α]$ 的条件，并给出了数域非单本的充分条件及特殊情形分析。
In 2021, Guyer and Mbirika gave two equivalent formulas that computed the greatest common divisor (GCD) of all sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In this current paper, we extend their results to the GCD of all sums of $k$ consecutive squares of these numbers. Denoting these GCD values by the symbol $\mathcal{G}_{G_0, G_1}^2\!(k)$, we prove $\mathcal{G}_{G_0, G_1}^2\!(k) = \gcd\left(G_k G_{k+1} - G_0 G_1,\; G_{k+1}…
Products of consecutive integers with unusual anatomy
本文研究了连续自然数区间中连乘积的性质，定义了bad、very bad和type F3区间，并给出了关于这些区间数量的渐进行为及相关方程的解的近似数量。
Topology of zero sets of polynomials with square discriminant
本研究针对满足特定封闭性的整数系数集合，证明了该集合系数的整系数多项式零点可由同集合系数、平方判别式的整系数多项式零点任意逼近，揭示这类多项式零点集复闭包的拓扑结构与判别式是否为平方无关，区别于伽罗瓦群的相关性质。
In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided by him to check $p$-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic $p$-rational fields have been shown to exist in recent years. In this article, for any integer $k \geq 1$, we build upon the existing work and prove the existence of infinitely many prime numbers $p$ for which the imaginary quadrat…
Sum of two repdigits a square
\emph{repdigit}是一个大于大于10的自然数，它的所有以10为基数的数字都是相同的。在本文中，我们发现所有的例子，两个repdigits添加到一个广场。这些证明引出了关于连续二次剩余和非剩余的有趣问题，并提供了椭圆曲线的一个初步应用。
Residue Class Patterns of Consecutive Primes
研究连续素数的剩余类模式，通过结合狄利克雷定理与Shiu定理，修改梅纳德 - 陶筛法并结合埃尔德什 - 兰金类型构造进行证明，得出在特定条件下关于长度为\(m\)的剩余类模式无限次出现的相关结论及数量估计。
When $s\ge k\ge 3$ and $n_1,\ldots ,n_k$ are large natural numbers, denote by $A_{s,k}(\mathbf n)$ the number of solutions in non-negative integers $\mathbf x$ to the system \[ x_1^j+\ldots +x_s^j=n_j\quad (1\le j\le k). \] Under appropriate local solubility conditions on $\mathbf n$, we obtain an asymptotic formula for $A_{s,k}(\mathbf n)$ when $s\ge k(k+1)$. This establishes a local-global principle in the Hilbert-Kamke problem at the convexity barrier. Our arguments involve minor arc estimates going beyond square-root cancellation.
On Bhargava's heuristics for $\mathbf{GL}_2(\mathbb{F}_p)$-number fields and the number of elliptic curves of bounded conductor
我们提出了一个新的模型，用于计数$\mathbf{GL}_2(\mathbb{F}_p)$-数域具有相同的本地属性的模$p$-伽罗瓦表示的分裂字段与椭圆曲线上的有理数。我们解释了这个新的模型和Bhargava的局部到全局算法如何计算$\mathbf{GL}_2(\mathbb{F}_p)$-数域都揭示了估计平方自由导体$N < X.$的有理数上的椭圆曲线的数目的问题。新的模型预测了比局部到全局算法明显更多的具有所需局部性质的$\mathbf{GL}_2(\mathbb{F}_p)$-数域的存在。当$N$具有大量不同的素因子时，由于Atkin-Lehner对合引起的退化，全局模型。我们描述了支持新模型的计算证据。
Hecke modules associated to certain even ternary lattices of half-discriminant $\mathbf{N^2}$
该研究针对半判别式为$N^2$的定偶三元格对应的Hecke模，借助Watson型变换序列，建立其与经典模形式空间的Jacquet-Langlands型显式对应，为相关数论问题提供了新的研究路径。
Primes in the intervals between primes squared
连续素数平方之间的短间隔集具有令人愉快的--但似乎未被利用的--性质，即每个间隔$s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$都被$k$第一个素数完全筛选。在这里，我们利用这一基本特征，并提出证据的猜想，即$π_k \sim |s_k|/ \log p_{k+1}^2$，其中$π_k$是素数在$s_k$的数量;或更严格的，即$y=x^{1/2}$是必要的和充分的素数定理是有效的长度为$y$的间隔。此外，我们提出并证实，素数计数函数$π(x)$是最好的理解为相关的随机变量$π_k$的总和。在此假设下，我们推导出$π(p_{k+1}^2)=\sum_{j=1}^k π_j$的理论方差，由此我们推测$|π({x})-\textrm{li}(x)| =O(\sqrt{\textrm{li}(x)})$。从我们的研究中出现的观点是，连续素数之间的间隔平方持有进一步理解素数分布的关键;如所证明的，这种观点也建立了有利于黎曼假设的强有力的支持。
Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass
本文研究奇数无平方因子水平 SK 新形式回拉的谱分解，将投影平方联系于 GL(3)×GL(2) 中心 L-值，并利用 CFKRS 启发法给出 L2-质量公式及非零性条件。
When $k$ and $s$ are natural numbers and $\mathbf h\in \mathbb Z^k$, denote by $J_{s,k}(X;\mathbf h)$ the number of integral solutions of the system \[ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\le j\le k), \] with $1\le x_i,y_i\le X$. When $s<k(k+1)/2$ and $(h_1,\ldots ,h_{k-1})\ne {\mathbf 0}$, Brandes and Hughes have shown that $J_{s,k}(X;\mathbf h)=o(X^s)$. In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov's mean value theorem, we obtain an asymptotic formula for $J_{s,k}(X;\mathbf h)$ in the critical case $s=k(k+1…
When $\mathbf h\in \mathbb Z^3$, denote by $B(X;\mathbf h)$ the number of integral solutions to the system \[ \sum_{i=1}^6(x_i^j-y_i^j)=h_j\quad (1\le j\le 3), \] with $1\le x_i,y_i\le X$ $(1\le i\le 6)$. When $h_1\ne 0$ and appropriate local solubility conditions on $\mathbf h$ are met, we obtain an asymptotic formula for $B(X;\mathbf h)$, thereby establishing a subconvex local-global principle in the inhomogeneous cubic Vinogradov system. We obtain similar conclusions also when $h_1=0$, $h_2\ne 0$ and $X$ is sufficiently large in terms of $h_2$. Our arguments involve minor arc estimates goin…
On symmetric square values of quadratic polynomials
证明了不存在一个非平方二次多项式的整数系数和一个对称轴，其中采取平方值的N个连续整数N=7或N >= 9。相反，如果N <= 6或N=8，则有无穷多个。
Spacing statistics for a point scatterer on the cubic three-torus
研究立方三维环面上点散射体的新特征值间距统计，证明其收敛于独立于参数的分布，并刻画了小间隙渐近形式。
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