There exist many results about the Diophantine equation $(q^n-1)/(q-1)=y^m$, where $m\ge 2$ and $n\ge 3$. In this paper, we suppose that $m=1$, $n$ is an odd integer and $q$ a power of a prime number. Also let $y$ be an integer such that the number of prime divisors of $y-1$ is less than or equal to $3$. Then we solve completely the Diophantine equation $(q^n-1)/(q-1)=y$ for infinitely many values of $y$. This result finds frequent applications in the theory of finite groups.

In this paper we complete the solution to the equation w+x+y = z, where w, x, y, and z are positive integers and wxyz has the form 2r 3s 5t, with r, s, and t non negative integers. Here we consider the case 1 < w ≤ x ≤ y, the remaining case having been dealt with in our paper: On the Diophantine equation 1+ X + Y = Z, Rocky Mountain J. of Math. This work extends earlier work of the authors in the field of exponential Diophantine equations.

In this paper, we find all integer solutions $(x,y,n,a,b,c)$ of the equation in the title for non-negative integers $a,b$ and $c$ under the condition that the integers $x$ and $y$ are relatively prime and $n\ge 3$. The proof depends on the famous primitive divisor theorem due to Bilu, Hanrot and Voutier and the computational techniques on some elliptic curves.

We find all the solutions of the Diophantine equation $x²+2^{\alpha }{13}^{\beta }=yⁿ$ in positive integers x,y,α,β,n ≥ 3 with x and y coprime.

In this paper, we find all solutions of the Diophantine equation $x^2+2^{\alpha }{5}^{\beta }{17}^{\gamma }=y^n$ in positive integers $x,y\ge 1$, $\alpha ,\beta ,\gamma ,n\ge 3$ with $gcd(x,y)=1$.

Consider the equation in the title in unknown integers $(x,y,k,l,n)$ with $x\ge 1$, $y>1$, $n\ge 3$, $k\ge 0$, $l\ge 0$ and $gcd(x,y)=1$. Under the above conditions we give all solutions of the title equation (see Theorem 1).

In this study, we determine when the Diophantine equation $x^2-kxy+y^2-2^n=0$ has an infinite number of positive integer solutions $x$ and $y$ for $0\le n\le 10.$ Moreover, we give all positive integer solutions of the same equation for $0\le n\le 10$ in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation $x^2-kxy+y^2-2^n=0$.

We attack the equation of the title using a Frey curve, Ribet’s level-lowering theorem and a method due to Darmon and Merel. We are able to determine all the solutions in pairwise coprime integers $x,y,z$ if $p\ge 7$ is prime and $k\ge 2$. From this we deduce some results about special cases of this equation that have been studied in the literature. In particular, we are able to combine our result with previous results of Arif and Abu Muriefah, and those of Cohn to obtain a complete solution for the equation $x^2+2^k=y^n$ for...