Let $c$ be a square-free positive integer and $p$ a prime satisfying $p\nmid c$. Let $h(-c)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-c})$. In this paper, we consider the Diophantine equation $$cx^2+p^{2m}=4y^n,~~x,y\geq 1, m\geq 0, n\geq 3, \gcd(x,y)=1, \gcd(n,2h(-c))=1,$$ and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.
Given odd, coprime integers $a$, $b$ ($a>0$), we consider the Diophantine equation $ax^2+b^{2l}=4y^n$, $x, y\in\Bbb Z$, $l \in \Bbb N$, $n$ odd prime, $\gcd(x,y)=1$. We completely solve the above Diophantine equation for $a\in\{7,11,19,43,67,163\}$, and $b$ a power of an odd prime, under the conditions $2^{n-1}b^l\not\equiv \pm 1(\mod a)$ and $\gcd(n,b)=1$. For other square-free integers $a>3$ and $b$ a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers $x$, $y$ with ($\gcd(x,y)=1$), $l\in\mathbb{N}$ and all odd primes $n>3$, satisfying $2^{n-…
University of Colombo（科伦坡大学）；Washington and Lee University（华盛顿与李大学）；University of Southern California（南加州大学）；University of California, Los Angeles（加州大学洛杉矶分校）；Williams College（威廉姆斯学院）
Suppose that $x$ is odd, $n\geq7$ and $p\notin\{2,5\}$ are primes. In this paper, we prove that the Diophantine equations $x^{2}\pm5^αp^{n}=y^{n}$ have no solutions in positive integers $α,x,y$ with $gcd(x,y)=1$.
Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $α$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(α)/\mathbb Q$ is Galois. We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type $$ |F(x, y)| = tp^k $$ in integers $(x, y, t)$ such that $\gcd(x, y) = 1$ and $1 \leq t \leq (p^k)^λ$ does not exceed $24$. Here $λ= λ(d)$ is a certain positive, monotonously increasing function that approaches one as $d$ tends to infinity. We also prove that, for every su…
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).
We investigate the solvability of the Diophantine equation in the title, where $d>1$ is a square-free integer, $p, q$ are distinct odd primes and $x,y,a,b$ are unknown positive integers with $\gcd(x,y)=1$. We describe all the integer solutions of this equation, and then use the main finding to deduce some results concerning the integers solutions of some of its variants. The methods adopted here are elementary in nature and are primarily based on the existence of the primitive divisors of certain Lehmer numbers.
Given $n\in N$ and $x,γ\in R$, let \begin{equation*} ||γ-nx||^\prime=\min\{|γ-nx+m|:m\in Z, \gcd (n,m)=1\}, \end{equation*} %where $(n,m)$ is the largest common divisor of $n$ and $m$. Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number $α$ and almost every $γ\in R$, \begin{equation*} \liminf_{n\to \infty}n||γ-nα||^{\prime}=0 \end{equation*} and that there exists $C>0$, such that for all $α\in R\backslash Q$ and $γ\in [0,1)$ , \begin{equation*} \liminf_{n\to \infty}n||γ-nα||^{\prime} < C. \end{equation*} We prove the first conjecture and …
The title equation is completely solved in integers $(n,x,y,a,b)$, where $n\geq 3$, $\gcd(x,y)=1$ and $a,b\geq 0$. The most difficult stage of the resolution is the explicit resolution of a quintic Thue-Mahler equation. Since it is for the first time -to the best of our knowledge- that such an equation is solved in the literature, we make a detailed presentation of the resolution; this gives our paper also an expository character.
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…
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 $$r\equiv 2(mod 4), m\equiv 3(mod 4), m>\max\…
In this paper, we consider the equation $(a^n-2^{m})(b^n-2^{m})=x^2$. By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the above equation has only finitely many solutions $n,x$ if a and b are different fixed positive integers. We solve the above equation when $m=1$ and $(a,b)=(2,10),(4,100),(10,58),(3,45)$. Moreover, we show that $(a^2-2)(b^2-2)=x^2$ has no solution n,x if 2|n and gcd$(a,b)=1$. We also give a conjecture which says that the equation $(2^2-2)((2P_k)^n-2)=x^2$ has only the solution $(n,x)=(2,Q_k)$, where $k>3$ is odd and $P_k,Q_k$ …
We completely solve the Diophantine equation $x^2+2^k11^\ell19^m=y^n$ in integers $x,y\geq 1;~ k,\ell, m\geq 0~$ and $n\geq 3$ with $\gcd(x,y)=1$, except the case $2\mid k, 2\nmid \ell m$ and $5\mid n$. We use this result to recover some earlier results in the same direction.
We formulate an exponential Diophantine equation, which is is some sense one order higher that Fermat's Last Theorem. We also give three examples of solutions to this exponential Diophantine equation and formulate a conjecture.
If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell$, with $\operatorname{gcd}(n,d)=1$ and $\ell \geq 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.
Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23 \pmod {24}$, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p\geqslant 5$. In this paper, we introduce new infinite Ramanujan type congruences modulo $2$ for $b_3(2n)$. They complement naturally the results of Keith-Zanello and Yao and involve primes in $\mathcal P=\{p \text{ prime …
We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.
In this paper, we solve Diophantine equation in the tittle in nonnegative integers m,n, and a. In order to prove our result, we use lower bounds for linear forms in logarithms and and a version of the Baker-Davenport reduction method in diophantine approximation.
In 1956, Je$\acute{s}$manowicz conjectured that, for positive integers $m$ and $n$ with $m>n, \, \gcd(m,\, n)=1$ and $m\not\equiv n\pmod{2}$, the exponential Diophantine equation $(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z$ has only the positive integer solution $(x,\,y,\, z)=(2,\,2,\,2)$. Recently, Ma and Chen \cite{MC17} proved the conjecture if $4\not|mn$ and $y\ge2$. In this paper, we present an elementary proof of the result of Ma and Chen \cite{MC17}.
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 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 prove effective joint equidistribution of several natural parameters associated to primitive vectors in $\mathbb{Z}^{n}$, as the norm of these vectors tends to infinity. These parameters include the direction, the orthogonal lattice, and the length of the shortest solution to the associated $\gcd$ equation. We show that the first two parameters equidistribute w.r.t. the Haar measure on the corresponding spaces, which are the unit sphere and the space of unimodular rank $n-1$ lattices in $\mathbb{R}^{n}$ respectively. The main novelty is the equidistribution of the shortest solutions to the …
\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.
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 $(x, y)=(1, 1)$.
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.
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}.
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 $a^2+b^2+c^2=abcf(a,b,c)$.
