Yongzhong Hu, Maohua Le
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 $(x,y,z)$.
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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.
On some ternary Diophantine equations of Signature $(p,p,k)$
本文综述了签名$(p,p,k)$的三元丢番图方程$Ax^n+By^n=cz^m$的研究，并完全解决了特定系数下的若干情形。
Additive Diophantine Equations involving S-Units, Factorials and Ternary Recurrences with repeated root
研究Cullen数C_n=n2^n+1与阶乘及S-单位构成的加法丢番图方程，证明其最大素因子趋于无穷，并求解特定方程。
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 $n^{5}<32\left(1+3(10^6)\right)^5$.
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。
On the Ternary Purely Exponential Diophantine Equation $(ak)^x+(bk)^y=((a+b)k)^z$ with Prime Powers $a$ and $b$
该研究结合初等数论方法与经典丢番图方程结果、Lucas序列新性质，证明了特定素数幂条件下一类三元纯指数丢番图方程仅有正整数解$(1,1,1)$，部分验证了已有猜想。
On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences
本文研究了 Padovan 数、Perrin 数和 Narayana's cows 数与基 $b$ 下 Thabit 数及 Williams 数相关的丢番图方程，确定了符合条件的数，并求出了 $2 \leq b \leq 10$ 范围内的所有解。
Purely exponential Diophantine equations with four terms of consecutive bases: contribution to Skolem's conjecture
本文研究具有四个连续底数项的纯指数丢番图方程，证明其仅存在两组正整数解，并对$n\ge4$的情况给出显式模数证明无解，为斯科伦关于纯指数丢番图方程局部-全局原理的经典问题提供了贡献。
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.
Solving an infinite number of purely exponential Diophantine equations with four terms
本文研究了具有连续基数的四项纯指数丢番图方程的无限族，证明了当 $n \equiv 3 \pmod{4}$ 时方程 $n^x+(n+1)^y+(n+2)^z=(n+3)^w$ 的所有正整数解，结合了初等同余论证、Baker 方法和基于 Skolem 猜想变体的算法。
Rational points close to non-singular algebraic curves
该研究聚焦于非奇异三元型对应的丢番图不等式解的密度问题，等价探讨接近非奇异平面代数曲线的有理点密度，为数论相关领域的有理点分布研究提供方向。
Handling some Diophantine equation via Euclidean algorithm and its application to purely exponential equations
本文通过经典和新方法研究三元指数丢番图方程，验证了R.Scar和R.Styer猜想在特定条件下成立，并应用于多项式-指数丢番图方程。
Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.
Perfect powers among Jacobsthal numbers
本文通过模方法和Bennett-Skinner框架，证明了Jacobsthal数中唯一的完美幂是平凡情况0和1。
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 formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields. As an application we show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly we construct and study "genuinely PDE" differential modular forms. As further applications we derive a Theorem of the Kernel and a Reciprocity Theore…
Small solutions of generic ternary quadratic congruences to general moduli
本文研究一般奇数模数下三元二次同余方程的小解，利用Burgess和Heath-Brown的特征和估计突破指数1/2障碍，在平均意义下对几乎所有系数得到解的渐近公式，并提出最小解大小的精细猜想。
On the solutions of $x^2= By^p+Cz^p$ and $2x^2= By^p+Cz^p$ over totally real fields
本文研究了全实域上素数指数丢番图方程 $x^2= By^p+Cz^p$ 及相关方程的解与非平凡本原解，并给出了该域的纯局部判据。
Values of ternary quadratic forms at integers and the Berry-Tabor conjecture for 3-tori
本文通过研究三次二次型在整数上的取值，验证了伯里-塔布猜想在三维扭体中的适用性，解决了特殊对角型在不同缩放下的分布问题。
Purely exponential parametrizations and their group-theoretic applications
本文通过提出纯指数参数化（PEP）理论，利用丢番图几何工具解决了线性群有界生成性质的若干长期未决问题，并证明了由半单元素有界生成的线性群必为实际上阿贝尔群。
General sharp bounds for the number of solutions to purely exponential equations with three terms
本文证明对任意固定c，三项纯指数方程a^x+b^y=c^z至多有一个解，仅有限多对a、b例外，推广了Miyazaki和Pink的Pillai型方程结果，并基于p-进方法与丢番图逼近理论。
At most one solution to $a^x + b^y = c^z$ for some ranges of $a$, $b$, $c$
本文证明了纯指数丢番图方程 $a^x+b^y=c^z$ 在特定范围内至多一个解，并推广到 Pillai 方程和 Jeśmanowicz 猜想，利用 Miyazaki-Pink 思想和 Scott 定理。
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