On equations (−1)αpx + (−1)β(2k(2p + 1))y = z2 with Sophie Germain prime p

Li, Yuan; Zhang, Jing; Liu, Baoxing

doi:10.2140/involve.2024.17.503

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

Yuan Li, Jing Zhang and Baoxing Liu

Vol. 17 (2024), No. 3, 503–518

DOI: 10.2140/involve.2024.17.503

Abstract

We consider the Diophantine equation ${(- 1)}^{α} p^{x} + {(- 1)}^{β} {(2^{k} (2 p + 1))}^{y} = z^{2}$ for Sophie Germain primes $p$ with $α, β \in {0, 1}$ , $α β = 0$ and $k \geq 0$ . First, for $p = 2$ , we solve three Diophantine equations of the form ${(- 1)}^{α} 2^{x} + {(- 1)}^{β} {(2^{k} 5)}^{y} = z^{2}$ using the Nagell–Ljunggren equation and elementary methods. Then we obtain all nonnegative integer solutions for the following four types of equations for odd Sophie Germain primes $p$ :

$p^{x} + {(2^{2 k + 1} (2 p + 1))}^{y} = z^{2}$ with $p \equiv 3, 5 (\mod 8)$ and $k \geq 0$ ,
$p^{x} + {(2^{2 k} (2 p + 1))}^{y} = z^{2}$ with $p \equiv 3 (\mod 8)$ and $k \geq 1$ ,
$p^{x} - {(2^{k} (2 p + 1))}^{y} = z^{2}$ with $p \equiv 3 (\mod 4)$ and $k \geq 0$ ,
$- p^{x} + {(2^{k} (2 p + 1))}^{y} = z^{2}$ with $p \equiv 1, 3, 5 (\mod 8)$ and $k \geq 1$ .

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Keywords

Catalan equation, exponential Diophantine equation, Legendre symbol, Nagell–Ljunggren equation, quadratic reciprocity law, Sophie Germain prime

Mathematical Subject Classification

Primary: 11A15, 11D61, 11D72, 14H52

Milestones

Received: 6 December 2022

Revised: 21 April 2023

Accepted: 29 April 2023

Published: 17 July 2024

Communicated by Nathan Kaplan

Authors

Yuan Li
Department of Mathematics Winston Salem State University Winston-Salem, NC United States

Jing Zhang
Department of Mathematics, Division of Science, Mathematics and Technology Governors State University University Park, IL United States

Baoxing Liu
Department of Mathematics Winston Salem State University Winston-Salem, NC United States

Vol. 17, No. 3, 2024