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.

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 $gcd(x,y)=1$.

We completely solve the Diophantine equations $x²+2^a{q}^b=yⁿ$ (for q = 17, 29, 41). We also determine all $C=p{₁}^{a₁}\cdots p_k^{a_k}$ and $C=2^{a₀}p{₁}^{a₁}\cdots p_k^{a_k}$, where $p₁,...,p_k$ are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).