Related papers: On a class of Lebesgue-Ramanujan-Nagell equations

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…

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…

We refine a recent result of Parsell on the values of the form $\lambda_1p_1 + \lambda_2p_2 + \mu_1 2^{m_1} + ...m + \mu_s 2^{m_s}, $ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $\lambda_1 / \lambda_2$ is…

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…

In the fall 2011 issue of the Journal'Mathematics and Computer Education', author Unal Hasan, in the one page article "Proof without Words", gives a purely geometric proof of the equality, arctan(1/3)+ arctan(1/7) = arctan(1/2) (1) (See…

In 2012, T. Miyazaki and A. Togb\'{e} gave all of the solutions of the Diophantine equations $(2am-1)^x+(2m)^y=(2am+1)^z$ and $b^x+2^y=(b+2)^z$ in positive integers $x,y,z,$ $a>1$ and $b\ge 5$ odd. In this paper, we propose a similar…

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…

Integer solutions of the diophantine equation $A^4+hB^4=C^4+hD^4$ are known for all positive integer values of $h < 1000$. While a solution of the aforementioned diophantine equation for any arbitrary positive integer value of $h$ is not…

In this paper we present a method to find the solutions to the Brocard-Ramanujan's problem $n! + 1 = m^2$.

The aim of this note is to show that given a positive integer $n \geq 5$, the positive integral solutions of the diophantine equation $4/n = 1/x + 1/y+1/z$ cannot have solution such that $x$ and $y$ are coprime with $xy < \sqrt{z/2}$. The…

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…

We study the Diophantine equation of type $U_n(x)=V_m(y)$, where $(U_n)_{n\geq 0}$ and $(V_m)_{m\geq 0}$ are polynomial power sums defined over a number field $K$. By applying the finiteness criterion of Bilu and Tichy, we show under…

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 investigate the Diophantine equation \[ (2^k - 1)(3^k - 1) = x^n \] and prove that it has no solutions in positive integers $k, x, n > 2$.

Let $p$ be a prime number with $p>3$, $p\equiv 3\pmod{4}$ and let $n$ be a positive integer. In this paper, we prove that the Diophantine equation $(5pn^{2}-1)^{x}+(p(p-5)n^{2}+1)^{y}=(pn)^{z}$ has only the positive integer solution…

We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…

Let $a$ and $b$ be two distinct fixed positive integers such that $\min \{a,b\}>1.$ First, we correct an oversight from \cite{X-Z}. Then, we show that the equation in the title with $b \equiv 3 \pmod 8$, $b$ prime and $a$ even has no…

This paper studies integer solutions to the Diophantine equation A+B=C in which none of A, B, C have a large prime factor. We set H(A, B,C) = max(|A|, |B|, |C|), and consider primitive solutions (gcd}(A, B, C)=1) having no prime factor p…

Here, we find all positive integer solutions of the Diophantine equation in the title, where $(\mathcal{U}_n)_{n\geqslant 0}$ is the generalized Lucas sequence $\mathcal{U}_0=0, \ \mathcal{U}_1=1$ and $\mathcal{U}_{n+1}=r \mathcal{U}_n +s…

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers…