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

TL;DR

This paper investigates specific Diophantine equations involving Sophie Germain primes, solving for solutions in particular cases and exploring the existence of such primes under various modular conditions, combining computational and theoretical methods.

Contribution

It provides explicit solutions for certain equations with Sophie Germain primes and discusses the existence of primes satisfying various modular conditions, advancing understanding of related Diophantine problems.

Findings

Solved equations for p=2 using elliptic curves and LMFDB database.

Established solutions for four types of equations with odd Sophie Germain primes.

Conjectured the infinitude of Sophie Germain primes in all residue classes modulo 8.

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