Related papers: On the equation $x^2-2^m=y^n$

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…

We investigate the Lebesgue--Nagell equation \begin{align*} x^2-2=y^p \end{align*} in integers $x,y,p$ with $p\geq 3$ an odd prime. A longstanding folklore conjecture asserts that the only solutions are the ``trivial'' ones with $y=-1$. We…

We consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c $ in nonnegative integers $x, y$ and integers $u, v \in \{0,1\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$. When…

In this paper we consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c$ in nonnegative integers $x, y$ and integers $u, v \in \{0,1\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$.…

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.

Consider the $n$th degree polynomial equation, $X^n+A_{n-1}X^{n-1}+...+A_1X+A_0=0$ over the ring of 2 by 2 complex matrices. If this equation has more than ${2n \choose 2}$ solutions, then it has infinitely many solutions. We show here that…

We determine all integers $n$ such that $n^2$ has at most three base-$q$ digits for $q \in \{2, 3, 4, 5, 8, 16 \}$. More generally, we show that all solutions to equations of the shape $$ Y^2 = t^2 + M \cdot q^m + N \cdot q^n, $$ where $q$…

The title equations were originally solved by making use of certain results on hypergeometric functions. Aside from these results, the classifications of the solutions uses very elementary arithmetic. The goal of this is to show that these…

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…

Let $\mathbb{N}$ be the set of all positive integers and let $a,\, b,\, c$ be nonzero integers such that $\gcd\left(a,\, b,\, c\right)=1$. In this paper, we prove the following three results: (1) the solvability of the matrix equation…

Let $n$ be a positive integer and consider the Diophantine equation of generalized Fermat type $x^2+y^{2n}=z^3$ in nonzero coprime integer unknowns $x,y,z$. Using methods of modular forms and Galois representations for approaching…

The object of this paper is to give a new proof of all the solutions of the Diophantine equation x^2+11^m=y^n; in positive integers x, y with odd m>1 and n>=3.

We estimate the number of solutions to $|y^2 - x^3| \leq X$ with $N \leq y \leq 2N$, in terms of both $N$ and $X$.

In this article, we study the following nonlinear Choquard equation with singular nonlinearity \begin{equation*} \quad -\De u = \la u^{-q} + \left( \int_{\Om}\frac{|u|^{2^*_{\mu}}}{|x-y|^{\mu}}\mathrm{d}y \right)|u|^{2^*_{\mu}-2}u, \quad…

Let $k$ be a fixed positive integer with $k>1$. In this paper, using various elementary methods in number theory, we give criteria under which the equation $x^2+(2k-1)^y=k^z$ has no positive integer solutions $(x,y,z)$ with $y\in\{3,5\}$.

For all odd positive integers $m$, we construct $\mu$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^3,$ with $\mu\in(m,m+1)$. For $m$ large, $\mu-m$ converges to $1$, so $\mu\neq m+\tfrac 1 2$. The restriction to odd…

In this paper we consider the Diophantine equation $x^2+q^{2m}=2y^p$ where $m,p,q,x,y$ are integer unknowns with $m>0,$ $p$ and $q$ are odd primes and $\gcd(x,y)=1.$ We prove that there are only finitely many solutions $(m,p,q,x,y)$ for…

We find all positive integer solutions in $x, y$ and $n$ of $x^2+19^{2k+1}=4y^{n}$ for any non-negative integer $k$.

This work determine the entire family of positive integer solutions of the diophantine equation. The solution is described in terms of $\frac{(m-1)(m+n-2)}{2} $ or $\frac{(m-1)(m+n-1)}{2}$ positive parameters depending on $n$ even or odd.…

Let $D_{1}$, $D_{2}$ be coprime odd integers with min$(D_{1},D_{2})>1$, and let $N(D_{1},D_{2})$ denote the number of positive integer solutions $(x, m, n)$ of the equation $D_{1}x^2+D_{2}^{m}=2^{n+2}$. In this paper, we prove that…