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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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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author Yuan Li
Torre Lloyd
Angel Clinton
author_facet Yuan Li
Torre Lloyd
Angel Clinton
author_sort Yuan Li
collection DOAJ
description 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).
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spelling doaj-art-85d499dda8cc4070ad4f5dbea0020ed12024-12-27T14:38:20ZengMDPI AGMathematics2227-73902024-12-011224402710.3390/math12244027On 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>Yuan Li0Torre Lloyd1Angel Clinton2Department of Mathematics, Winston-Salem State University, Winston-Salem, NC 27110, USADepartment of Mathematics, Winston-Salem State University, Winston-Salem, NC 27110, USADepartment of Mathematics, Winston-Salem State University, Winston-Salem, NC 27110, USAIn 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).https://www.mdpi.com/2227-7390/12/24/4027Catalan equationelliptic curve
spellingShingle Yuan Li
Torre Lloyd
Angel Clinton
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>
Mathematics
Catalan equation
elliptic curve
title 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>
title_full 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>
title_fullStr 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>
title_full_unstemmed 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>
title_short 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>
title_sort 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
topic Catalan equation
elliptic curve
url https://www.mdpi.com/2227-7390/12/24/4027
work_keys_str_mv AT yuanli ondiophantineequations2supixisup2supikisupipisupiyisupizisup2supand2supixisup2supikisup3supiyisupizisup2sup
AT torrelloyd ondiophantineequations2supixisup2supikisupipisupiyisupizisup2supand2supixisup2supikisup3supiyisupizisup2sup
AT angelclinton ondiophantineequations2supixisup2supikisupipisupiyisupizisup2supand2supixisup2supikisup3supiyisupizisup2sup

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