Diophantine problems related to the Omega constant

On certain diophantine equations related to triangular and tetrahedral numbers

In this paper we give solutions of certain diophantine equations related to triangular and tetrahedral numbers and propose several problems connected with these numbers. The material of this paper was presented in part at the 11th International Workshop for Young Mathematicians - NUMBER THEORY, Kraków, 14th-20th september 2008.

An exponential diophantine equation related to odd perfect numbers

We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give its application to odd perfect number problem.

Diophantine problems related to cyclic cubic and quartic fields

We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively.

On a Conjecture of Sun Zhi-Wei and Related Diophantine Equations

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.

On a Problem in Diophantine Approximation

We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.

Some Diophantine equations related to positive-rank elliptic curves

We give conditions on the rational numbers a,b,c which imply that there are infinitely many triples (x,y,z) of rational numbers such that x+y+z=a+b+c and xyz=abc. We do the same for the equations x+y+z=a+b+c and x^3+y^3+z^3=a^3+b^3+c^3. These results rely on exhibiting families of positive-rank elliptic curves.

The average number of solutions of the Diophantine equation U^2 + V^2 = W^3 and related arithmetic functions

This paper provides asymptotics with a sharp error term for the Dirichlet summatory function of a certain class of arithmetic functions. The result applies, e.g., to the sums over r^2(n) and r(n^3), where r(m) denotes the number of ways to write m as a sum of two squares of integers.

On certain Littlewood-like and Schmidt-like problems in inhomogeneous Diophantine approximations

We give several results related to inhomogeneous approximations to two real numbers and badly approximable numbers. Our results are related to classical theorems by A. Khintchine (1926) and to an original method invented by Y. Peres and W. Schlag (2001).

Complete solution of the diophantine equation $X^{2}+1=dY^{4}$ and a related family of quartic Thue equations

In this paper, we use the method of Thue and Siegel, based on explicit Pade approximations to algebraic functions, to completely solve a family of quartic Thue equations. From this result, we can also solve the diophantine equation in the title. We prove that this equation has at most one solution in positive integers when $d \geq 3$. Moreover, when such a solution exists, it is of the form $(u,\sqrt{v})$ where $(u,v)$ is the fundamental solution of $X^{2}+1=dY^{2}$.

On Kleinbock's Diophantine result

We give an elementary proof of a recent metrical Diophantine result by D. Kleinbock related to badly approximable vectors in affine subspaces.

乘法矩阵方程的可判定性及相关丢番图问题

Decidability of multiplicative matrix equations and related Diophantine problems

本文研究了代数数域上乘法矩阵方程的可判定性，通过有效方法分析丢番图问题，提出了解的大小和数量的显式界，并探讨了对称矩阵系统的相关计数问题。

On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

Let $r\ge 1$ be an integer and ${\bf U}:=(U_{n})_{n\ge 0} $ be the Lucas sequence given by $U_0=0$, $U_1=1, $ and $U_{n+2}=rU_{n+1}+U_n$, for all $ n\ge 0 $. In this paper, we show that there are no positive integers $r\ge 3,~x\ne 2,~n\ge 1$ such that $U_n^x+U_{n+1}^x$ is a member of ${\bf U}$.

关于与幂电路相关的丢番图问题

On the Diophantine problem related to power circuits

本文研究了与幂电路相关的丢番图问题，证明了在特定结构下该问题不可判定。

弱逼近、迪奥菲安指数和二维格点

Weak approximations, Diophantine exponents and two-dimensional lattices

本文研究二维格点的迪奥菲安指数及其相关弱统一逼近性质，利用连分数理论分析普通与统一迪奥菲安指数间的不等式关系，揭示两个实数无理数度量函数的行为关联。

An octic diophantine equation and related families of elliptic curves

We obtain two parametric solutions of the diophantine equation $ϕ(x_1, x_2, x_3)=ϕ(y_1, y_2, y_3)$ where $ϕ(x_1, x_2, x_3)$ is the octic form defined by $ϕ(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 - 2x_2^4x_3^4$. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank~$5$ over $\mathbb{Q}(t)$. We study one of the two families in some detail and determine a set of five free generators for the family.

A sextic diophantine chain and a related Mordell curve

In this paper we obtain parametric as well as numerical solutions of the sextic diophantine chain $ ϕ(x_1,\,y_1,\,z_1)=ϕ(x_2,\,y_2,\,z_2)=ϕ(x_3,\,y_3,\,z_3)=k$ where $ϕ(x,\,y,\,z)$ is a sextic form defined by $ϕ(x,\,y,\,z)$ $=x^6+y^6+z^6-2x^3y^3-2x^3z^3-2y^3z^3$ and $k$ is an integer. Each numerical solution of such a sextic chain yields, in general, nine rational points on the Mordell curve $y^2=x^3+k/4$. While all of these nine points are not independent in the group of rational points of the Mordell curve, we have constructed a parameterized family of Mordell curves of generic rank $\geq 6$…

与角平分线相关的丢番图方程及佩尔方程的解

Diophantine equation related to angle bisectors and solutions of Pell's equations

本文将寻找斜率均为有理数的两直线及其角平分线的问题转化为丢番图方程，并利用负佩尔方程的解给出了全部非平凡整数解，同时借助等直角边毕达哥拉斯三元组给出了有理数解公式。

On Diophantine approximations with positive integers: a remark to W.M.Schmidt's theorem

We prove a generalization of W.M. Schmidt's theorem related to the Diophantine approximations for a linear form of the type $α_1x_1+α_2x_2 +y$ with {\it positive} integers $x_1,x_2$.

A note on Arithmetic Diophantine series

We consider some asymptotic analysis for series related to the work of Hardy and Littlewood on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH.

关于分形的度量丢番图逼近

Metric Diophantine approximation on fractals

该研究探讨了缺失数字集中的无理数与多项式分母有理数的逼近问题，并试图证明一个针对大基数缺失数字集的Khintchine型收敛定理。

关于与有限域本原元相关的丢番图\(m\)元组

On Diophantine $m$-tuples related to primitive elements of finite fields

本文研究与有限域本原元相关的丢番图\(m\)元组，给出\(\mathbb{F}_q\)上\(\mathcal{P}\)-丢番图\(m\)元组数量\(N_m\)的渐近公式，并证明当\(q\)足够大时存在这样的\(m\)元组。

阿贝尔簇上的丢番图逼近；M. Waldschmidt的一个猜想

Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt

本文研究阿贝尔簇上的丢番图逼近，将Waldschmidt猜想等价转化为矩阵条件，提出较弱猜想并证明其上界，对秩1椭圆曲线获得弱型Dirichlet定理并证明猜想。

On primitive integer solutions of the Diophantine equation $t^2=G(x,y,z)$ and related results

In this paper we investigate Diophantine equations of the form $T^2=G(\overline{X}),\; \overline{X}=(X_{1},\ldots,X_{m})$, where $m=3$ or $m=4$ and $G$ is specific homogenous quintic form. First, we prove that if $F(x,y,z)=x^2+y^2+az^2+bxy+cyz+dxz\in\Z[x,y,z]$ and $(b-2,4a-d^2,d)\neq (0,0,0)$, then the Diophantine equation $t^2=nxyzF(x,y,z)$ has solution in polynomials $x, y, z, t$ with integer coefficients, without polynomial common factor of positive degree. In case $a=d=0, b=2$ we prove that there are infinitely many primitive integer solutions of the Diophantine equation under consideratio…

A Diophantine Frobenius problem related to Riemann surfaces

We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair $(p,q)$, $2<p<q$, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realized as a semi-regular $pq$-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order $pq$. The general $t$-di…

On $px^2 + q^{2n}= y^p$ and related Diophantine equations

The title equation, where $p>3$ is a prime number $\not\equiv 7 \pmod 8$, $q$ is an odd prime number and $x,y,n$ are positive integers with $x,y$ relatively prime, is studied. When $p\equiv 3\pmod 8$, we prove (Theorem 2.3) that there are no solutions. For $p\not\equiv 3\pmod 8$ the treatment of the equation turns out to be a difficult task. We focus our attention to $p=5$, by reason of an article by F. Abu Muriefah, published in this journal, vol. 128 (2008), 1670-1675. Our main result concerning this special equation is Theorem 1.1, whose proof is based on results around the Diophantine equa…

Inhomogeneous Diophantine approximation of some Hurwitzian numbers

We continue the work of Takao Komatsu by considering the inhomogeneous approximation constant L(θ,ϕ) for Hurwitzian numbers θ, and rationally related ϕ(r θ+m)/n in Q(θ) +Q. The current work uses a compactness theorem to relate such inhomogeneous constants to the homogeneous approximation constants. Among the new results are: a characterization of such pairs θ,ϕfor which L(θ,ϕ) is zero; consideration of small values of n^2 L(e^{2/s},ϕ); and the proof of a conjecture of Komatsu.

Hausdorff Dimension and Diophantine Approximation

We begin with a brief treatment of Hausdorff measure and Hausdorff dimension. We then explain some of the principal results in Diophantine approximation and the Hausdorff dimension of related sets, originating in the pioneering work of Vojtech Jarnik. We conclude with some applications of these results to the metrical structure of exceptional sets associated with some famous problems. It is not intended that all the recent developments be covered but they can be found in the references cited.

On discrepancy, intrinsic Diophantine approximation, and spectral gaps

In the present paper we establish bounds for the size of the spectral gap for actions of algebraic groups on certain homogeneous spaces. Our approach is based on estimating operator norms of suitable averaging operators, and we develop techniques for establishing both upper and lower bounds for such norms. We shall show that this analytic problem is closely related to the arithmetic problem of establishing bounds on the discrepancy of distribution for rational points on algebraic group varieties. As an application, we show how to establish an effective bound for property $τ$ of congruence subg…

On products of disjoint blocks of arithmetic progressions and related equations

In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erdős and Graham.

Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

In this paper, we associate an invariant $α_{x}(L)$ to an algebraic point $x$ on an algebraic variety $X$ with an ample line bundle $L$. The invariant $α$ measures how well $x$ can be approximated by rational points on $X$, with respect to the height function associated to $L$. We show that this invariant is closely related to the Seshadri constant $ε_{x}(L)$ measuring local positivity of $L$ at $x$, and in particular that Roth's theorem on $\mathbf{P}^1$ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

On decompositions of quadrinomials and related Diophantine equations

Let $A,B,C,D$ be rational numbers such that $ABC \neq 0$, and let $n_1>n_2>n_3>0$ be positive integers. We solve the equation $$ Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = f(g(x)),$$ in $f,g \in \mathbb{Q}[x]$. In sequel we use Bilu-Tichy method to prove finitness of integral solutions of the equations $$ Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = Ey^{m_1}+Fy^{m_2}+Gy^{m_3}+H, $$ where $A,B,C,D,E,F,G,H$ are rational numbers $ABCEFG \neq 0$ and $n_1>n_2>n_3>0$, $m_1>m_2>m_3>0$, $\gcd(n_1,n_2,n_3) = \gcd(m_1,m_2,m_3)=1$ and $n_1,m_1 \geq 9$. And the equation $$ A_1x^{n_1}+A_2x^{n_2}+\ldots+A_l x^{n_l} + A_{l+1} = Ey^{m…

关于一对丢番图方程的问题

Problems Regarding a Pair of Diophantine Equations

本文研究了两个互质正整数参数下的丢番图方程解的唯一性，并总结了相关最新研究成果及未来研究问题。

The Repulsion Motif in Diophantine Equations

Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a non-singular plane cubic equation has only finitely many integral solutions. Examples show that simple equations can have inordinately large integral solutions in comparison to the size of their coefficients. A conjecture of Hall attempts to ameliorate this by bounding the size of integral solutions simply in terms of the coefficients of the defining equation. It turns out that a simi…

Rational points near manifolds and metric Diophantine approximation

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindzuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarnik type divergence theorems are established for arbitrary analytic non-degenerate manifolds regardless of their dimension. The key to establishing these results is the study o…

A note on weighted badly approximable linear forms

We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give a affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

On two exponents of approximation related to a real number and its square

For any irrational real number xi, let lambda(xi) denote the supremum of all real numbers lambda such that, for each sufficiently large X, the inequalities |x_0| < X, |x_0*xi-x_1| < X^{-lambda} and |x_0*xi^2-x_2| < X^{-lambda} admit a solution in integers x_0, x_1 and x_2 not all zero, and let omega(xi) denote the supremum of all real numbers omega such that, for each sufficiently large X, the dual inequalities |x_0+x_1*xi+x_2*xi^2| < X^{-omega}, |x_1| < X and |x_2| < X admit a solution in integers x_0, x_1 and x_2 not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that t…

Appearance of Balancing and related number sequences in steady state probabilities of some Markov chains

Balancing and Lucas-balancing numbers are solutions of a Diophantine equation and satisfy a second order homogeneous recurrence relation. Interestingly, these numbers can be seen as numerators and denominators in the steady state probabilities of a class of transition probability matrices of Markov chains. An identity relating the balancing numbers and the silver ratio can be obtained as a byproduct.

Finiteness Problems in Diophantine Geometry

This survey contains an exposition of ideas and results related to Faltings' proof of the conjectures of Shafarevich, Tate and Mordell. This paper originally appeared in 1986 as an Appendix to the Russian translation of Serge Lang, "Fundamentals of Diophantine Geometry" (Springer Verlag, 1983) published by "Mir", Moscow (MR0854670, 88a:11054). A history of the publication of the Appendix is briefly described by Lang in Section 4 of his paper "Mordell's review, Siegel's letter to Mordell, Diophantine geometry, and 20th century mathematics" that was published (in 1995) simultaneously in Notices …

Triples which are $D(n)$-sets for several $n$'s

For a nonzero integer $n$, a set of distinct nonzero integers $\{a_1,a_2,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq i<j\leq m$, is called a Diophantine $m$-tuple with the property $D(n)$ or simply $D(n)$-set. $D(1)$-sets are known as simply Diophantine $m$-tuples. Such sets were first studied by Diophantus of Alexandria, and since then by many authors. It is natural to ask if there exists a Diophantine $m$-tuple ($D(1)$-set) which is also a $D(n)$-set for some $n\neq 1$. This question was raised by Kihel and Kihel in 2001. They conjectured that there are no Diophanti…

定量丢番图逼近度量理论中的有效结果

Effective Results in The Metric Theory of Quantitative Diophantine Approximation

本文将丢番图逼近度量理论中依赖隐式常数的渐近定量化为有效结果，在测度至多 δ 的例外集外显式控制常数，并推出 Schmidt 定理、Koukoulopoulos–Maynard 定理及正规数等结果。

Triangles with two given integral sides

We study some Diophantine problems related to triangles with two given integral sides. We solve two problems posed by Zoltán Bertalan and we also provide some generalization.

双向图、整二次型与若干丢番图方程

Bidirected graphs, integral quadratic forms and some Diophantine equations

本文研究双向图关联的非负整二次型（关联型），刻画其 Dynkin 型分类并引入 $\mathbb{C}$ 型，同时将图形框架用于研究根、丢番图方程解及 gentle 代数的 Euler 二次型。

Perfect Powers: Pillai's works and their developments

After a short introduction to Pillai's work on Diophantine questions, we quote some later developments and we discuss related open problems.

A singular K3 surface related to sums of consecutive cubes

We study the surface arising from the diophantine equation $m^3+(m+1)^3+...+(m+k-1)^3=l^2$. It turns out that this is a $K3$ surface with Picard number 20. We stduy its aritmetic properties in detail. We construct elliptic fibrations on it, and we find a parametric solution to the original equation. Also, we determine the Hasse-Weil zeta function of the surface over $Q$.

Figurate primes and Hilbert's 8th problem

In this paper, by using the theory of elliptic curves, we discuss several Diophantine equations related with the so-called figurate primes. Meanwhile, we raise several conjectures related with figurate primes and Hilbert's 8th problem, including Goldbach's conjecture, twin primes conjecture and Catalan's conjecture as well.

Diophantine Approximation with Prime Restriction in Real Quadratic Number Fields

The distribution of $αp$ modulo one, where $p$ runs over the rational primes and $α$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $ν>0$ one can establish the infinitude of primes $p$ satisfying $||αp||\le p^{-ν}$. The latest record in this regard is Kaisa Matomäki's landmark result $ν=1/3-\varepsilon$ which presents the limit of currently known technology. Recently, Glyn Harman, and, jointly, Marc Technau and the first-named author, investigated the same problem in the context of imaginary quadratic fields. Glyn Harman obtained an analog…

Power values of sums of certain products of consecutive integers and related results

Let $n$ be a non-negative integer and put $p_{n}(x)=\prod_{i=0}^{n}(x+i)$. In the first part of the paper, for given $n$, we study the existence of integer solutions of the Diophantine equation $$ y^m=p_{n}(x)+\sum_{i=1}^{k}p_{a_{i}}(x), $$ where $m\in\N_{\geq 2}$ and $a_{1}<a_{2}<\ldots <a_{k}<n$. This equation can be considered as a generalization of the Erdős-Selfridge Diophantine equation $y^m=p_{n}(x)$. We present some general finiteness results concerning the integer solutions of the above equation. In particular, if $n\geq 2$ with $a_{1}\geq 2$, then our equation has only finitely many …

关于庞加莱的一个例子与带克罗内克序列的和

On an example by Poincaré and sums with Kronecker sequence

受相关近期研究启发，本文整合庞加莱关于带克罗内克序列的连续周期函数和的例子及推广，给出Kochergin结果的改进高维表述，证明普适连续函数不存在，并首次记录涉及三类丢番图指数的光滑性结果。

Limit theorems for toral translations

We discuss some classical and recent results and open problems on the statistical behavior of ergodic sums above toral translations, and their applications to Diophantine approximations and to ergodic properties of systems related to quasi-periodic dynamics such as skew products, cylindrical cascades and special flows.