Hairong Bai, Elif Kızıldere, Gökhan Soydan, Pingzhi Yuan
Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z},\ n\geq 2,\ xyz\neq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.
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Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developments involving smooth Weyl sums. Next, arithmetic variants of classical inequalities of Bessel and Cauchy-Schwarz are discussed. Finally, some emerging connections between the circle method and arithmetic geometry are mentioned.
We formulate an exponential Diophantine equation, which is is some sense one order higher that Fermat's Last Theorem. We also give three examples of solutions to this exponential Diophantine equation and formulate a conjecture.
We propose a conjecture, similar to Skolem's conjecture, on a Hasse-type principle for exponential diophantine equations. We prove that in a sense the principle is valid for "almost all" equations. Based upon this we propose a general method for the solution of exponential diophantine equations. Using a generalization of a result of Erdős, Pomerance and Schmutz concerning Carmichael's $λ$ function, we can make our search systematic for certain moduli needed in the method.
In this paper, we find all solutions of the exponential Diophantine equation $B_{n+1}^x-B_n^x=B_m$ in positive integer variables $(m, n, x)$, where $B_k$ is the $k$-th term of the Balancing sequence.
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 $a^x + b^y = c^z$.
In this paper, we explicitly find all solutions of the title Diophantine equation, using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker-Davenport reduction method in Diophantine approximation, due to Dujella and Pethö. This paper extends the previous work of \cite{Patel}.
In this paper, we prove that there is no x>=4 such that the difference of x-th powers of two consecutive Fibonacci numbers greater than 0 is a Lucas number.
We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.
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 $(x,y,z)$.
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.
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).
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 $(x,y,z)$.
On the exponential Diophantine equation $x^2+p^mq^n=2y^p$
本研究结合Lehmer序列本原因子与初等数论方法，探讨含奇素数$p,q$、正整数$x,y,m,n$的指数丢番图方程$x^2+p^mq^n=2y^p$的可解性。
This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.
We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution.
In this paper we obtain three undecidable results for exponential diophantine equations over the field $\mathbb Q$ of rational numbers. For example, we prove that there is no algorithm to decide the solvability of a general exponential diophantine equation $F(x_1,\ldots,x_{8})=0$ over $\mathbb Q$ with eight unknowns.
An Exponential Diophantine equation $x^2+3^α 113^β=y^{\mathfrak{n}}$
本文针对满足互素条件的指数丢番图方程 $x^2+3^α113^β=y^{\frak{n}}$，开展其在正整数变量、非负指数及不小于3的整数指数下的全部解的确定研究。
Families of Unit Equations and Exponential Diophantine Problems via Integral Points
本文通过Ru-Vojta定理和Huang-Levin-Xiao不等式推广研究了投影簇上整数点的分布，证明了两类一参数单位方程解集的退化性，并扩展了GCD估计以获得特定指数迪菲antine方程和q进制表示中数字分布的新结果。
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 ${\bf U}$.
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。
In 2002, F. Luca and G. Walsh solved the Diophantine equation in the title for all pairs (a,b) such that 1<a<b<101 with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation in the title. It is proved that the equation (a^n-1)(b^n-1)=x^2 has no solutions if a,b have opposite parity and n>4 with 2|n. Also, we solved (a^n-1)(b^n-1)=x^2 for the pairs (a,b)=(2,50),(4,49),(12,45),(13,76),(20,77),(28,49), and (45,100). Lastly, we show that when b is even, the equation (a^n-1)(b^(2n)a^n-1)=x^2 has no solutions n,x.
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 $(x,y,z)=(1,1,2)$.
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。
In this paper, we consider the exponential Diophantine equation $a^{x}+b^{y}=c^{z},$ where $a, b, c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{r}, r\in Z^{+}, 2\mid r$ with $b$ even. That is $$a=\mid Re(m+n\sqrt{-1})^{r}\mid, b=\mid Im(m+n\sqrt{-1})^{r}\mid, c=m^{2}+n^{2},$$ where $m, n$ are positive integers with $m>n, m-n\equiv1(mod 2),$ gcd$(m, n)=1.$ $(x, y, z)= (2, 2, r)$ is called the trivial solution of the equation. In this paper we prove that the equation has no nontrivial solutions in positive integers $x, y, z$ when $$r\equiv 2(mod 4), m\equiv 3(mod 4), m>\max\…
On the Exponential Diophantine Equation $(a^n-1)(b^n-1)=x^2$
本文纠正了先前文献的疏漏，并证明在特定条件下（$b$ 为模 8 余 3 的素数，$a$ 为偶数）指数丢番图方程无正整数解，推广了 Szalay 的结果。
Let $A$, $B$ be fixed positive integers such that $\min\{A,B\} > 1$, $\gcd(A,B) = 1$ and $AB \equiv 0 \bmod 2$, and let $n$ be a positive integer with $n>1$. In this paper, using a deep result on the existence of primitive divisors of Lucas numbers due to Y. Bilu, G. Hanrot and P. M. Voutier \cite{BHV}, we prove that if $A > 8 B^3$, then the equation $(*) \quad (A^2 n)^x + (B^2 n)^y = ((A^2 + B^2)n)^z$ has no positive integer solutions $(x,y,z)$ with $x > z > y$. Combining the above conclusion with some existing results, we can deduce that if $A >8 B^3$ and $B \equiv 2 \bmod 4$, then (*) has o…
On the exponential Diophantine equation $(a^n+1)(b^n+1)=x^2$
研究 Szalay 2000 年工作衍生的一类方程 $(a^n+1)(b^n+1)=x^2$，证明当 $a$ 和 $b$ 是同一整数 $t>1$ 的不同幂时，方程只有一组正整数解，并在互质且 $n$ 为偶数的条件下给出解的完整描述。
A number alpha in R is diophantine if it is not well approximable by rationals, i.e. for some C, nu>0 and any relatively prime p, q in Z we have |alpha q -p|>C q^{-1-\vu}. It is well-known and easy to prove that almost every alpha in R is diophantine. In this paper we address a noncommutative version of the question about diophantine properties. Consider a pair A,B in SO(3) and for each n in Z_+ take all possible words in A, A^{-1}, B, and B^{-1} of length n, i.e. for a multiindex I=(i_1,j_1,..., i_s,j_s) put |I|=\sum_{k=1}^s (|i_k|+|j_k|) and W_{A,B}(n)={W_{I}(A,B)= A^{i_1}B^{j_1}... A^{i_s}B…
\noindent In this article, we determine all the integers $c$ having at least two representations as difference between two linear recurrent sequences. This is a variant of the Pillai's equation. This equation is an exponential Diophantine equation. The proof of our main theorem uses lower bounds for linear forms of logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.
In this paper, the theory of elliptic curves is used for finding the solutions of the quartic Diophantine equation $X^4+Y^4=2(U^4+V^4)$ Keywords: Diophantine equation, Elliptic curve, Congruent number
In this work, we accomplish three goals. First, we determine the entire family of positive integer solutions to the three- variable Diophantine equation, xy=z^2; for n=2,3,4,5,6. For n=2, we obtain a 3-parameter family of solutions; for n=3, a 5-parameter of solutions; likewise for n=4. For n=5, a 7-parameter family of solutions; and likewise for n=6. See Theorems 2 through 6 respectively. The second goal of this paper, is determining all the positive integer solutions of xyz=w^2. This is done in Theorem7; the solution set is described in terms of six independent parameters. Finally, in Theore…
Exponential Diophantine approximation and symbolic dynamics
本文扩展了交织乘性Markoff-Lagrange谱与符号动力学的关键公式，运用复分析进行证明，并将新方法推广至可能含重根的多项式，导出了谱上的若干预期推论。
We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.
On the exponential Diophantine equation ${\displaystyle p\cdot 3^{x}+p^{y}=z^2}$ with $p$ a prime number
本文利用模算术性质研究素数 $p$ 下的指数丢番图方程 $p\cdot 3^x+p^y=z^2$，分类给出不同同余情形的唯一解或无穷解集，并讨论无解情形与开放问题。
In this paper, we solve Diophantine equation in the tittle in nonnegative integers m,n, and a. In order to prove our result, we use lower bounds for linear forms in logarithms and and a version of the Baker-Davenport reduction method in diophantine approximation.
One of the main open problems regarding decidability of the existential theory of rings is the analogue of Hilbert's Tenth Problem (HTP) for the ring of entire holomorphic functions in one variable. In the direction of a negative solution, we prove unsolvability of HTP for rings of exponential polynomials. This provides the first known case of HTP for a ring of entire holomorphic functions in one variable strictly containing the polynomials. The technique of proof consists of an interaction between Arithmetic, Analysis, Logic, and Functional Transcendence.
In this paper we suggest an implementation of Runge's method for solving Diophantine equations satisfying Runge's condition. In this implementation we avoid the use of Puiseux series and algebraic coefficients.
In this paper, we consider the equation $(a^n-2^{m})(b^n-2^{m})=x^2$. By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the above equation has only finitely many solutions $n,x$ if a and b are different fixed positive integers. We solve the above equation when $m=1$ and $(a,b)=(2,10),(4,100),(10,58),(3,45)$. Moreover, we show that $(a^2-2)(b^2-2)=x^2$ has no solution n,x if 2|n and gcd$(a,b)=1$. We also give a conjecture which says that the equation $(2^2-2)((2P_k)^n-2)=x^2$ has only the solution $(n,x)=(2,Q_k)$, where $k>3$ is odd and $P_k,Q_k$ …
For every positive integer $n$, an infinite family of positive integral solutions of the diophantine equation $x^n - y^n = z^{n+1}$ is constructed.
The paper shows that the asymptotic density of solutions of Diophantine equations or systems of the natural numbers is 0. The author provides estimation methods and estimates number, density and probability of k- tuples $<x_1,...x_k>$ to be the solution of the algebraic equations of the first, second and higher orders in two or more variables, non-algebraic Diophantine equations and systems of Diophantine equations in the domain of the natural numbers. The estimate for the number of positive integer solutions of the second-order Diophantine equations in two, three or more variables is geometri…
We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various Diophantine equations. We also give many equivalent statements of Mahler's generalization of the fundamental theorem of Thue. In particular, we show that the theorem of Thue--Mahler for degree $3$ implies the theorem of Thue--Mahler for arbitrary degree $\ge3$, and we relate it with a theorem of Siegel on the rational integral points of the projective line $¶^1…
In this paper we obtain a parametric solution of the hitherto unsolved diophantine equation $(x_1^5+x_2^5)(x_3^5+x_4^5)=(y_1^5+y_2^5)(y_3^5+y_4^5)$. Further, we show, using elliptic curves, that there exist infinitely many parametric solutions of the aforementioned diophantine equation, and they can be effectively computed.
In this paper we show a way to generalize the linear Diophantine equation a1x1+a2x2+...+anxn=d . We deal with the nonlinear Diophantine equation det|A X|=+-d , which generalizes the linear one, and we give a necessary and sufficient condition for its solvabiliy. We show how the linear equation can be considered as a particular case of the nonlinear equation.
The paper assesses the top number of integer solutions for algebraic Diophantine Thue diagonal equation of the degree $n \geq 2$ and number of variables $k > 2$ and equations with explicit variable in the case when the coefficients of the equation are of the opposite signs. The author found integer conversions that maintain the asymptotic behavior of the number of integer solutions of algebraic Diophantine equation in the case of the conversion equation to diagonal form. The paper considers the estimation of the number of integer solutions for some types of algebraic Diophantine equations with…
The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f(x) in Z[x] monic and q1 ,...., qm in Z, we study the conditions for which the Diophantine equatio (y + q1 )(y + q2 ) .... (y + qm ) = f(x) has finitely many solutions in integers. Also assuming ABC Conjecture, we study the conditions for finiteness of integer solutions of the Diophantine equation f(x) = g(y).
In this paper, we obtain new results on the integers solutions X, Y, Z of the diophantine equation X^t+Y^t=BZ^t for a rationnal integer B and a prime number t verifying some conditions explained in the paper.
In an earlier paper, Tatong and Suvarnamani explores the Diophantine equation $p^x + p^y = z^2$ for a prime number $p$. In that paper they find some solutions to the equation for $p=2, 3$. In this paper, we look at a general version of this equation and solve it completely.
We show that the diophantine equation $n^\ell+(n+1)^\ell + ...+ (n+k)^\ell=(n+k+1)^\ell+ ...+ (n+2k)^\ell$ has no solutions in positive integers $k,n \ge 1$ for all $\ell \ge 3$.
In this paper, we prove a theorem about the integer solutions to the Diophantine equation $x^{4}-q^{4}=py^{r}$, extending previous work of K.Gy\H ory, and F.Luca and A.Togbe, and of the author.
Let $f(x)=x^{2}(x^{2}-1)(x^{2}-2)(x^{2}-3).$ We prove that the Diophantine equation $ f(x)=2f(y)$ has no solutions in positive integers $x$ and $y$, except $(x, y)=(1, 1)$.
This is an expository lecture on the subject of the title delivered at the Park-IAS mathematical institute in Princeton (July, 2000).
Let p, c be distinct odd primes, and l \ge 2 an integer. We find sufficient conditions for the Diophantine equation cy^l=(x^p-1)/(x-1) not to have integer solutions
The modular approach to Diophantine equations over totally real fields
本文综述了将Wiles证明Fermat大定理的模方法推广至全实数域上丢番图方程时所面临的新障碍，并调查了该领域现有的方法与结果。
In this paper we obtain several parametric solutions of the quartic diophantine equation $(x_1^4+x_2^4)(y_1^4+y_2^4)=z_1^4+z_2^4$. We also show how infinitely many parametric solutions of this equation may be obtained by using elliptic curves.
We propose an equivalent condition for the Markov triples, which was mentioned by H. Rademacher essentially. As an application, we study the solvability of the Diophantine equation $a^2+b^2+c^2=abcf(a,b,c)$.
Starting from Ritt's classical theorems, we give a survey of results in functional decomposition of polynomials and of applications in Diophantine equations. This includes sufficient conditions for the indecomposability of polynomials, the study of decompositions of lacunary polynomials and the finiteness criterion for the equations of type f(x) = g(y).
This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients without a free member via the circle method of Hardy and Littlewood. Author found the upper bound of the number of natural solutions of inhomogeneous algebraic Diophantine diagonal equations with explicit variable. He developed a method in the paper, which allows you to perform the low estimate of the number of natural (integer) solutions of algebraic Diophantine equation with integer coefficients. Author obtained a lower estim…
In this paper, we prove an asymptotic formula for the average number of solutions to the Diophantine equation $axy-x-y=n$ in which $a$ is fixed and and $n$ varies.
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。
In this article, we consider the equation x^2+19^{m}=y^n, n>2, m>0. We find the solutions of the title equation for not only 2 \mid m but also 2\notdividesm.
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