  In this paper, we consider the Diophantine equation
$(-1)^{\alpha}p^x+(-1)^{\beta}(2^k(2p+1))^y=z^2$ for Sophie Germain prime $p$
with $\alpha, \beta \in\{0,1\}$, $\alpha\beta=0$ and $k\geq 0$. First, for
$p=2$, we solve three Diophantine equations
$(-1)^{\alpha}2^x+(-1)^{\beta}(2^{k} 5)^y=z^2$ by using Nagell-Lijunggren
Equation and the database LMFDB of elliptic curve $y^2=x^3+ax+b$ over
$\mathbb{Q}$. Then we obtain all non-negative integer solutions for the
following four types of equations for odd Sophie Germain prime $p$:
  i) $p^x+(2^{2k+1}(2p+1))^y=z^2$ with $p\equiv 3, 5 \pmod 8$ and $k\geq 0$;
  ii) $p^x+(2^{2k}(2p+1))^y=z^2$ with $p\equiv 3 \pmod 8$ and $k\geq 1$;
  iii) $p^x-(2^{k}(2p+1))^y=z^2$ with $p\equiv 3 \pmod 4$ and $k\geq 0$;
  iv) $-p^x+(2^{k}(2p+1))^y=z^2$ with $p\equiv 1, 3, 5 \pmod 8$ and $k\geq 1$;
  For each type of the equations, we show the existences of such prime $p$.
Since it was conjectured that there exist infinitely many Sophie Germain primes
in literature, it is reasonable to conjecture that there exist infinite Sophie
Germain primes $p$ such that $p\equiv k \pmod 8$ for any $k\in\{1,3,5,7\}$.
