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 $k\ge 0$. 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 $p\equiv 3,5(\mathrm{mod}8)$ and $k\ge 0$; ii) $p^x+(2^{2k}(2p+1){)}^y=z^2$ with $p\equiv 3(\mathrm{mod}8)$ and $k\ge 1$; iii) $p^x-(2^k(2p+1){)}^y=z^2$ with $p\equiv 3(\mathrm{mod}4)$ and $k\ge 0$; iv) $-p^x+(2^k(2p+1){)}^y=z^2$ with $p\equiv 1,3,5(\mathrm{mod}8)$ and $k\ge 1$; 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 $p\equiv k(\mathrm{mod}8)$ for any $k\in \{1,3,5,7\}$.