On Diophantine Equations 2<sup><i>x</i></sup> ± (2<sup><i>k</i></sup><i>p</i>)<sup><i>y</i></sup> = <i>z</i><sup>2</sup> and −2<sup><i>x</i></sup> + (2<sup><i>k</i></sup>3)<sup><i>y</i></sup> = <i>z</i><sup>2</sup>

On Diophantine Equations 2<sup><i>x</i></sup> ± (2<sup><i>k</i></sup><i>p</i>)<sup><i>y</i></sup> = <i>z</i><sup>2</sup> and −2<sup><i>x</i></sup> + (2<sup><i>k</i></sup>3)<sup><i>y</i></sup> = <i>z</i><sup>2</sup>

In this paper, we solve three Diophantine equations: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mi>x</mi></msup><mo>±</mo><msup>&...

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Bibliographic Details
Main Authors:	Yuan Li, Torre Lloyd, Angel Clinton
Format:	Article
Language:	English
Published:	MDPI AG 2024-12-01
Series:	Mathematics
Subjects:	Catalan equation elliptic curve
Online Access:	https://www.mdpi.com/2227-7390/12/24/4027
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Summary:	In this paper, we solve three Diophantine equations: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mi>x</mi></msup><mo>±</mo><msup><mrow><mo>(</mo><msup><mn>2</mn><mi>k</mi></msup><mi>p</mi><mo>)</mo></mrow><mi>y</mi></msup><mo>=</mo><msup><mi>z</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msup><mn>2</mn><mi>x</mi></msup><mo>+</mo><msup><mrow><mo>(</mo><msup><mn>2</mn><mi>k</mi></msup><mn>3</mn><mo>)</mo></mrow><mi>y</mi></msup><mo>=</mo><msup><mi>z</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> and prime <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>≡</mo><mo>±</mo><mn>3</mn></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="4.44443pt"></mspace><mo>(</mo><mo form="prefix">mod</mo><mspace width="0.277778em"></mspace><mn>8</mn><mo>)</mo></mrow></semantics></math></inline-formula>. We obtain all the non-negative integer solutions by using elementary methods and the database of elliptic curves in “The L-functions and modular forms database” (LMFDB).
ISSN:	2227-7390