On equations $(-1)^{\alpha}p^x+(-1)^{\beta}(2^k(2p+1))^y=z^2$ with Sophie Germain prime $p$

Abstract

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 $k\in\{1,3,5,7\}$.