An equivalent condition for the Markov triples and the Diophantine equation $a^2+b^2+c^2=abcf(a,b,c)$

关于形式为{a^k, b, c}的k次幂丢番图三元组

On $k$-th power Diophantine triples of the form $\{a^k, b, c\}$

本文证明了对于k≥3且1<a^k<b<c的情况，形式为{a^k, b, c}的k次幂丢番图三元组不存在。

On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields

The possible torsion groups of elliptic curves induced by Diophantine triples over quadratic fields, which do not appear over Q, are Z/2Z x Z/10Z, Z/2Z x Z/12Z and Z/4Z x Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these torsion groups.

Diophantine triples in linear recurrence sequences of Pisot type

The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness of Diophantine triples in such sequences. Whilst the case of binary recurrence sequences is almost completely solved, not much was known about recurrence sequences of larger order, except for very specialized generalizations of the Fibonacci sequence. Now, we will prove that any linear recurrence sequence with the Pisot property contains only …

Strong rational Diophantine D(q)-triples

We show that for infinitely many square-free integers q there exist infinitely many triples of rational numbers {a, b, c} such that a^2 + q, b^2 + q, c^2 + q, ab + q, ac + q and bc + q are squares of rational numbers.

Diophantine triples with the property $D(n)$ for distinct $n$

We prove that for every integer $n$, there exist infinitely many $D(n)$-triples which are also $D(t)$-triples for $t\in\mathbb{Z}$ with $n\ne t$. We also prove that there are infinitely many triples with the property $D(-1)$ in $\mathbb{Z}[i]$ which are also $D(n)$-triple in $\mathbb{Z}[i]$ for two distinct $n$'s other than $n = -1$ and these triples are not equivalent to any triple with the property $D(1)$.

Finiteness results for Diophantine triples with repdigit values

Let $g\ge 2$ be an integer and $\mathcal R_g\subset \mathbb N$ be the set of repdigits in base $g$. Let $\mathcal D_g$ be the set of Diophantine triples with values in $\mathcal R_g$; that is, $\mathcal D_g$ is the set of all triples $(a,b,c)\in \mathbb N^3$ with $c<b<a$ such that $ab+1,ac+1$ and $ab+1$ lie in the set $\mathcal R_g$. In this paper, we prove effective finitness results for the set $\mathcal D_g$.

Two-parameter families of uniquely extendable Diophantine triples

Let A,K be positive integers and u=-2,-1,1 or 2. The main contribution of the paper is a proof that each of the D(u^2)-triples {K,A^2K+2uA,(A+1)^2K+2u(A+1)} has unique extension to a D(u^2)-quadruple.

Diophantine triples and construction of high-rank elliptic curves

Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial torsion points. By specialization, we obtain an example of elliptic curve over Q with torsion group Z/2Z * Z/2Z whose rank is equal 7.

On the extensions of the Diophantine triples in Gaussian integers

A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine triple $\{k-1, k+1, 16k^3-4k\}$ in Gaussian integers $\mathbb{Z}[i]$ to a Diophantine quadruple. Similar one-parameter family, $\{k-1, k+1, 4k\}$, was studied in Franušić's previous paper, where it was shown that the extension to a Diophantine quadruple is unique (with an element $16k^3-4k$). The family of the triples of the same form $\{k-1, k+1, 16k^3-4k\}$ was already studied in rational integers.…

On Mordell-Weil groups of elliptic curves induced by Diophantine triples

We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.

关于实数域上多项式环\(\mathbb{R}[X]\)中的奇异丢番图三元组

On exotic Diophantine triples in $\mathbb{R}[X]$

研究实数域上多项式环\(\mathbb{R}[X]\)中奇异丢番图三元组问题，通过证明得出若\(a,b,c\)都不等于\(1\)且至少有一个非常数元素时不存在该三元组，即含非常数元素的此类三元组必含\(1\)。

关于三角数的丢番图对和三数组

On Diophantine pairs and triples of triangular numbers

研究非零整数a下三角数的D(a)丢番图对和三数组，证明若三角数属于D(a)对，则可扩展为无穷多个D(a)三数组，并确定存在和不存在此类对的整数a的无限族。

Elliptic curves induced by Diophantine triples

Given a Diophantine triple $\{c_1(t),c_2(t),c_3(t)\}$, the elliptic curve over Q(t) induced by this triple, i.e. $y^2=(c_1(t) x+1) (c_2(t) x+1) (c_3(t) x+1)$, can have as torsion group one of the non-cyclic groups in Mazur's theorem, i.e. Z/2Z x Z/2Z, Z/2Z x Z/4Z, Z/2Z x Z/6Z or Z/2Z x Z/8Z. In this paper we present results concerning the rank over Q(t) of these curves improving in some of the cases the previously known results.

由正则丢番图三元组诱导的椭圆曲线的算术

Arithmetic of elliptic curves induced by regular Diophantine triples

本文研究正则丢番图三元组诱导的椭圆曲线的算术，明确其挠子群结构，给出二次域上获3阶点的判定准则并验证特定族无此情况，还分析另一类D(-k²)三元组诱导曲线的挠与秩。

Diophantine三元组{2, b, c}的可扩展性

The extensibility of the Diophantine triple $\{2, b, c\}$

本文研究了Diophantine三元组{2, b, c}的可扩展性，证明了其无法扩展为非规则四元组，并给出了相关结论。

关于正则高次幂有理数迪菲安特三元组

On Regular Higher Power Rational Diophantine Triples

本文研究了正则高次幂有理数迪菲安特三元组，构造了非平凡的无限族，并证明了当k=4时存在无限多个正元素三元组，还简要讨论了k=6和k=8的情况。

Diophantine triples with values in $k$-generalized Fibonacci sequences

We show that if $k\ge 2$ is an integer and $(F_n^{(k)})_{n\ge 0}$ is the sequence of $k$-generalized Fibonacci numbers, then there are only finitely many triples of positive integers $1<a<b<c$ such that $ab+1,~ac+1,~bc+1$ are all members of $\{F_n^{(k)}: n\ge 1\}$. This generalizes a previous result (cf. arXiv:1508.07760) where the statement for $k=3$ was proved. The result is ineffective since it is based on Schmidt's subspace theorem.

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…

与丢番图 $D(4)$-三元组相关的欧几里得三角形

Euclidean Triangles Associated with Diophantine $D(4)$-triples

本文研究了与丢番图 $D(4)$-三元组相关的欧几里得三角形，给出了非退化条件、数量判据、角类型、相似必全等结论，并对特定族获得了海伦三角形的佩尔方程刻画。

关于某些$D(9)$和$D(64)$丢番图三元组

On certain $D(9)$ and $D(64)$ Diophantine triples

研究关于$D(9)$和$D(64)$丢番图三元组，推广刻画特定$D(l^2)$-丢番图三元组对应佩尔方程解的结果，利用该结果及代数数对数线性形式的界，对特定形式的$D(9)$和$D(64)$-丢番图三元组进行分类。

Optimality of two inequalities for exponents of Diophantine approximation

We investigate two inequalities of Bugeaud and Laurent, each involving triples of classical exponents of Diophantine approximation associated to $\ux\in\mathbb{R}^n$. We provide a complete description of parameter triples that admit equality for suitable $\ux$, which turns out rather surprising. For $n=2$ our results agree with work of Laurent. Moreover, we establish lower bounds for the Hausdorff and packing dimensions of the involved $\ux$, and in special cases we can show they are sharp. Proofs are based on the variational principle in parametric geometry of numbers, we enclose sketches of …

关于丢番图三元组对应的椭圆曲线上整数点的杜杰拉（Dujella）猜想的反例

Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple

本文针对丢番图三元组对应椭圆曲线上整数点的杜杰拉猜想，构造无穷多组反例，证明存在x≠-1的整数点使得三个因子均非平方数，否定了该猜想的弱化问题。

线性递推中丢番图三元组的多项式变体

A Polynomial Variant of Diophantine Triples in Linear Recurrences

本文研究线性递推序列中满足乘积加固定多项式仍落在序列内的多项式三元组，证明在主导根条件且相关项非完全平方时此类三元组仅有有限多个。

与丢番图三元组$x^2+y^2=z^r$相关的Frobenius数（扩展版）

Frobenius numbers associated with Diophantine triples of $x^2+y^2=z^r$ (extended version)

本研究推导了与丢番图方程$x^2+y^2=z^r$相关三元组的$p$-Frobenius数显式公式，明确了其定义，补充了$r=2$时该数已有结论的背景，拓展了相关数论问题的研究结果。

A Family of Cubic Diophantine Equations and 4-Chains

In a simple integer chain, if $u_{i-1}$, $u_i$, and $u_{i+1}$ are three consecutive terms of the chain, and the pair $(u_{i-1}, u_i)$ has a certain property, then the next pair $(u_i, u_{i+1})$ also has the same property. We extend the idea of a simple chain to an $n$-chain in which $n$ is a positive integer and if a pair $(u_{i-1}, u_{i})$ has a certain property, then the $n$th next pair $(u_{\pm n+i-1}, u_{\pm n+i})$ also has the same property. In this case, we call $(u_{i-1}, u_{i}, u_{i+1})$ and $(u_{\pm n+i-1}, u_{\pm n+i}, u_{\pm n+i+1})$ matching triples. We use $4$-chains to study a fa…

构造可判定的丢番图方程

Creating Decidable Diophantine Equations

本文推广Matiyasevich的论证，提出一种生成无穷多个解可由线性递推完全刻画的丢番图方程的方法，并构造出唯一整数根为Tribonacci数连续三元组的整系数多项式。

Diophantine triples and K3 surfaces

A Diophantine $m$-tuple with elements in the field $K$ is a set of $m$ non-zero (distinct) elements of $K$ with the property that the product of any two distinct elements is one less than a square in $K$. Let $X: (x^2-1)(y^2-1)(z^2-1)=k^2,$ be a threefold. Its $K$-rational points parametrize Diophantine triples over $K$ such that the product of the elements of the triple that corresponds to the point $(x,y,z,k)\in X(K)$ is equal to $k$. We denote by $\overline{X}$ the projective closure of $X$ and for a fixed $k$ by $X_k$ a variety defined by the same equation as $X$. We prove that the variety…

Diophantine approximation by special primes

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.

A Note on Terai's Conjecture Concerning the Exponential Diophantine Equation x^{2}+b^{y}=c^{z}

Let (a,b,c) be a primitive Pythagorean triple, i.e., a^{2}+b^{2}=c^{2} with gcd(a,b,c)=1, a even and b odd. Terai's conjecture says that the Diophantine equation x^{2}+b^{y}=c^{z} has only the positive integer solutions (x,y,z)=(a,2,2). In this study, we prove that Terai's conjecture is true when b is a product of two primes and c=5(mod8).

Diophantine approximation with one prime of the form $p=x^2+y^2+1$

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 $p_3=x^2 + y^2 +1$.

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.

Notes on the Diophantine Equation A^4+aB^4=C^4+aD^4

A new formulation of the subject equation is presented. Several parametric and semi-parametric solutions are derived. The parametric solution for a=-1 was originally presented in 1972, but never published. A computer-generated version was found by Zajta and published in 1983. The pencil-and-paper version is presented for the record, along with many other examples, including eight coincidental triplet solutions.

Diophantine approximation by Piatetski-Shapiro primes

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 equation that generates all integral Apollonian Gaskets

A remarkably simple Diophantine quadratic equation is known to generate all Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also occurrences of Pythagorean triples in such gaskets is discussed.

Diophantine approximation in angular domains

Let $α$ and $β$ be real numbers such that $1$, $α$ and $β$ are linearly independent over $\mathbb{Q}$. A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+αx_1+βx_2| < \max\{|x_1|,|x_2|\}^{-2}$. In 1976, W. M. Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $γ\cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb{Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + αx_1 + βx_2| \max\{|x_1|,|x_2|\}^γ$ is arbitrarily small. Although Sc…

丢番图元组的又一个 $S$-单位变体

Yet another $S$-unit variant of Diophantine tuples

证明满足任意两数之积为给定多项式在正整数$S$-单位处取值的整数三元组仅有有限个，基于Corvaja和Zannier的结果及Schmidt子空间定理。

Polynomial parametrization of the solutions of Diophantine equations of genus 0

Let f in Z[X,Y,Z] be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by f=0 has a function field isomorphic to the rational function field Q(t). We show that all integral solutions of the Diophantine equation f=0 (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a single triple of polynomials with integer coefficients.

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。

A note on Diophantine systems involving three symmetric polynomials

Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $σ_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for each $1\leq i\leq n$ the system of Diophantine equations \begin{equation*} σ_{i}(\bar{X}_{2n})=a, \quad σ_{2n-i}(\bar{X}_{2n})=b, \quad σ_{2n}(\bar{X}_{2n})=c \end{equation*} has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. …

Explicit Bounds For The Diophantine Equation A!B! = C!

A nontrivial solution of the equation A!B! = C! is a triple of positive integers (A, B, C) with A $\le$ B $\le$ C -- 2. It is conjectured that the only nontrivial solution is (6, 7, 10), and this conjecture has been checked up to C = 10 6. Several estimates on the relative size of the parameters are known, such as the one given by Erd{ö}s C -- B $\le$ 5 log log C, or the one given by Bhat and Ramachandra C --B $\le$ (1/ log 2+o(1)) log log C. We check the conjecture for B $\le$ 10 3000 and give better explicit bounds such as C -- B $\le$ log log(B+1) log 2 -- 0.8803.

混合幂Piatetski-Shapiro素数的丢番图逼近

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。

Markoff Equation and Nilpotent Matrices

A triple (a,b,c) of positive integers is called a Markoff triple iff it satisfies the diophantine equation a2 + b2 + c2 = abc . Recasting the Markoff tree, whose vertices are Markoff triples, in the framework of intergral upper triangular 3x3 matrices, it will be shown that the largest member of such a triple determines the other two uniquely. This answers a question which has been open for almost 100 years.

On the Markoff Equation

A triple (a, b, c) of positive integers is called a Markoff triple iff it satisfies the Diophantine equation a2+b2+c2=abc . Recasting the Markoff tree, whose vertices are Markoff triples, in the framework of integral upper triangular 3x3 matrices, it will be shown that the largest member of such a triple determines the other two uniquely. This answers a question which has been open for 100 years. The solution of this problem will be obtained in the course of a broader investigation of the Markoff equation by means of 3x3 matrices.

涉及不同素数幂的丢番图不等式

A Diophantine inequality involving different powers of primes of the form $[n^c]$

研究在素数形式[n^c]下，不同幂的素数满足的丢番图不等式，证明存在无限多个素数三元组满足特定不等式条件。

丢番图分析与辫群${\bf B}_3$

Diophantine analysis and the Braid group ${\bf B}_3$

研究辫群${\bf B}_3$及其约化Burau表示相关特征曲面中素数三元组分布，通过定义特征多项式，证明该曲面中素数三元组出现频率高于周围格点，揭示群表示理论与解析数论的联系。

三角数的丢番图m元组

Diophantine m-tuples of Triangular Numbers

本文研究由三角数构成的D(n)丢番图元组，证明不存在D(1)三角四元组，提出生成D(1)三角三元组的算法，利用广义佩尔方程设计高效判定算法，还得出相关数论结论并推进了判据研究。

A Note on Je{ś}manowicz' Conjecture for Non-primitive Pythagorean Triples

Let $(a, b, c)$ be a primitive Pythagorean triple parameterized as $a=u^2-v^2,\ b=2uv,\ c=u^2+v^2$,\ where $u>v>0$ are co-prime and not of the same parity. In 1956, L. Je{ś}manowicz conjectured that for any positive integer $n$, the Diophantine equation $(an)^x+(bn)^y=(cn)^z$ has only the positive integer solution $(x,y,z)=(2,2,2)$. In this connection we call a positive integer solution $(x,y,z)\ne (2,2,2)$ with $n>1$ exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a un…

Class Numbers and Pell's Equation $x^2 + 105y^2 = z^2$

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently, Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In arXiv:2112.03663 we generalized these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a giv…

Conway's light on the shadow of Mordell

Recently Valentin Ovsienko introduced a ``shadow" version of the celebrated Markov triples as the solutions of certain version of Markov equation over dual numbers. We will discuss similar question for the Mordell Diophantine equation $$ X^2+Y^2+Z^2=2XYZ+1. $$ We will see that the shadows of the special solution $(1,1,1)$ can be described using the original Conway topograph of the values of binary quadratic forms. The shadows of other Mordell triples will be written explicitly in terms of the corresponding solutions of Pell's equation. Their growth along the paths on the Conway topograph is de…

Continued fractions and orderings on the Markov numbers

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbe…

Convexity and Aigner's Conjectures

Markov numbers are integers that appear in triples which are solutions of a Diophantine equation, the so-called Markov cubic $$x^2 + y^2 + z^2 - 3x y z = 0.$$ A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. One can associate to each a positive rational number a Markov number in a natural way. We give a new unified proof of certain conjectures from Martin Aigner's book, Markov's Theorem and 100 Years of the Uniqueness Conjecture. Our proof relies on a relationship be…