Related papers: On a conjecture on exponential Diophantine equatio…

In 2016 Izadi and Nabardi (b) showed (4-2-4) has infinitely many integer solutions. They used a specific congruent number elliptic curve.In 2019 Janfada and Nabardi,item C, showed that a necessary condition for n to have an integral…

Let $k,l\geq2$ be fixed integers. In this paper, firstly, we prove that all solutions of the equation $(x+1)^{k}+(x+2)^{k}+...+(lx)^{k}=y^{n}$ in integers $x,y,n$ with $x,y\geq1, n\geq2$ satisfy $n<C_{1}$ where $C_{1}=C_{1}(l,k)$ is an…

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p…

Let $1<c<\frac{26088036}{12301745},c\not=2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R\in (N,2N]$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c-R\big|<\log^{-1}N…

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…

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…

In this article, I study and solve the exponential Diophantine equation $M_p^{x} + (M_q + 1)^{y}= (lz)^2$ where $M_p$ and $M_q$ are Mersenne primes, $l$ is a prime number, and $x,y$, and $z$ are non-negative integers. Several illustrations…

In this paper, we give a specific way of describing positive integer solutions of a Diophantine equation $(x+y)^2+(y+z)^2+(z+x)^2=12xyz$ and introduce a generalized cluster pattern behind it.

We prove that the Diophantine equation (a^2 c x^k - 1)(b^2 c y^k - 1) = (abc z^k - 1)^2 has no solutions in positive integers with x, y, z > 1, k \geq 7 and a^2x^k \neq b^2y^k.

Crux Mathematicorum with Mathematical Mayhem, is a problem solving journal published by the Canadian Mathematical Society. In the March 2010 issue(see reference[1]) ,the following problem was proposed:Determine all positive integers a,b,…

In this paper we study the Diophantine equation $x^{4}-q^{4}=py^{5},$ with the following conditions: $p$ and $q$ are different prime natural numbers, $y$ is not divisible with $p$, $p\equiv3$ (mod20), $q\equiv4$ (mod5), $\overline{p}$ is a…

In this paper, we investigate solutions to the Diophantine equation $ A a^p + B b^p = C c^3 $ over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general…

We give a survey on some results covering the last 60 years concerning Je\'smanowicz' conjecture. Moreover, we conclude the survey with a new result by showing that the special Diophantine equation $$(20k)^x+(99k)^y=(101k)^z$$ has no…

Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$…

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.

Using the modular method, we study solutions to the Diophantine equation $$Aa^p+Bb^p=Cc^2$$ over number fields. We first prove an asymptotic result for general number fields satisfying an appropriate $S$-unit condition by assuming some…

Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $\alpha$ with $\alpha>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n…

In this paper we will explore the solutions to the diophantine equation in the Erd\H{o}s-Straus conjecture. For a prime $p$ we are discussing the relationship between the values $x,y,z \in \mathbb{N}$ so that $$ \frac{4}{p} = \frac{1}{x} +…

In this paper, we solve the simultaneous Diophantine equations(SDE) x_1^u+...+x_n^u=k(y_1^u+...+y_{n/k}); u=1,3, where n >3, and k< n, is a divisor of n , and obtain nontrivial parametric solution for them. Furthermore we present a method…

Let $u_n$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, $w$ a fixed non-zero integer and $p_1,p_2,\dots,p_s$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $u_n+u_m=w…