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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
##### Abstract

We consider the Diophantine equation ${\left(-1\right)}^{\alpha }{p}^{x}+{\left(-1\right)}^{\beta }{\left({2}^{k}\left(2p+1\right)\right)}^{y}={z}^{2}$ for Sophie Germain primes $p$ with $\alpha ,\beta \in \left\{0,1\right\}$, $\alpha \beta =0$ and $k\ge 0$. First, for $p=2$, we solve three Diophantine equations of the form ${\left(-1\right)}^{\alpha }{2}^{x}+{\left(-1\right)}^{\beta }{\left({2}^{k}5\right)}^{y}={z}^{2}$ 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 $p$:

1. ${p}^{x}+{\left({2}^{2k+1}\left(2p+1\right)\right)}^{y}={z}^{2}$ with $p\equiv 3,5\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}8\right)$ and $k\ge 0$,

2. ${p}^{x}+{\left({2}^{2k}\left(2p+1\right)\right)}^{y}={z}^{2}$ with $p\equiv 3\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}8\right)$ and $k\ge 1$,

3. ${p}^{x}-{\left({2}^{k}\left(2p+1\right)\right)}^{y}={z}^{2}$ with $p\equiv 3\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}4\right)$ and $k\ge 0$,

4. $-{p}^{x}+{\left({2}^{k}\left(2p+1\right)\right)}^{y}={z}^{2}$ with $p\equiv 1,3,5\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}8\right)$ and $k\ge 1$.

##### 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
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