Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions

Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions

Diophantine equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munderover><mstyle mathsize="90%" displaystyle="true"><mo>∏</mo></mstyle><mrow>...

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Autor principal: Petr Karlovsky
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
Diophantine equation
parametric solution
Lagrange multiplier
sums and products
upper bound
lower bound
Mathematics
QA1-939
Acceso en línea:https://doaj.org/article/84716101e415455d887f20b0defa87f3
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id oai:doaj.org-article:84716101e415455d887f20b0defa87f3
record_format dspace
institution DOAJ
collection DOAJ
language EN
topic Diophantine equation
parametric solution
Lagrange multiplier
sums and products
upper bound
lower bound
Mathematics
QA1-939
spellingShingle Diophantine equation
parametric solution
Lagrange multiplier
sums and products
upper bound
lower bound
Mathematics
QA1-939
Petr Karlovsky
Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions
description Diophantine equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munderover><mstyle mathsize="90%" displaystyle="true"><mo>∏</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mi>F</mi><munderover><mstyle mathsize="90%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>,</mo><mi>F</mi><mo>∈</mo><msup><mo>ℤ</mo><mo>+</mo></msup></mrow></semantics></math></inline-formula> associate the products and sums of <i>n</i> natural numbers. Only special cases have been studied so far. Here, we provide new parametric solutions depending on <i>F</i> and the divisors of <i>F</i> or <i>F</i><sup>2</sup>. One of these solutions shows that the equation of any degree with any <i>F</i> is solvable. For <i>n</i> = 2, exactly two solutions exist if and only if <i>F</i> is a prime. These solutions are (2<i>F</i>, 2<i>F</i>) and (<i>F</i> + 1, <i>F</i>(<i>F</i> + 1)). We generalize an upper bound for the sum of solution terms from <i>n</i> = 3 established by Crilly and Fletcher in 2015 to any <i>n</i> to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mi>F</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>F</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and determine a lower bound to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mroot><mrow><msup><mi>n</mi><mi>n</mi></msup><mi>F</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mroot></mrow></semantics></math></inline-formula>. Confining the solutions to <i>n</i>-tuples consisting of distinct terms, equations of the 4th degree with any <i>F</i> are solvable but equations of the 5th to 9th degree are not. An upper bound for the sum of terms of distinct-term solutions is conjectured to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mi>F</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>[</mo><mrow><mi>F</mi><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>!</mo><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>/</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>!</mo></mrow></semantics></math></inline-formula>. The conjecture is supported by computation, which also indicates that the upper bound equals the largest sum of solution terms if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>=</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mi>k</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>!</mo><mo>−</mo><mn>1</mn><mrow><mo>,</mo><mo> </mo></mrow><mi>k</mi><mo>∈</mo><msup><mo>ℤ</mo><mo>+</mo></msup></mrow></semantics></math></inline-formula>. Computation provides further insights into the relationships between <i>F</i> and the sum of terms of distinct-term solutions.
format article
author Petr Karlovsky
author_facet Petr Karlovsky
author_sort Petr Karlovsky
title Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions
title_short Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions
title_full Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions
title_fullStr Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions
title_full_unstemmed Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions
title_sort diophantine equations relating sums and products of positive integers: computation-aided study of parametric solutions, bounds, and distinct-term solutions
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/84716101e415455d887f20b0defa87f3
work_keys_str_mv AT petrkarlovsky diophantineequationsrelatingsumsandproductsofpositiveintegerscomputationaidedstudyofparametricsolutionsboundsanddistincttermsolutions
_version_ 1718431895149084672
spelling oai:doaj.org-article:84716101e415455d887f20b0defa87f32021-11-11T18:19:14ZDiophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions10.3390/math92127792227-7390https://doaj.org/article/84716101e415455d887f20b0defa87f32021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/21/2779https://doaj.org/toc/2227-7390Diophantine equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munderover><mstyle mathsize="90%" displaystyle="true"><mo>∏</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mi>F</mi><munderover><mstyle mathsize="90%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>,</mo><mi>F</mi><mo>∈</mo><msup><mo>ℤ</mo><mo>+</mo></msup></mrow></semantics></math></inline-formula> associate the products and sums of <i>n</i> natural numbers. Only special cases have been studied so far. Here, we provide new parametric solutions depending on <i>F</i> and the divisors of <i>F</i> or <i>F</i><sup>2</sup>. One of these solutions shows that the equation of any degree with any <i>F</i> is solvable. For <i>n</i> = 2, exactly two solutions exist if and only if <i>F</i> is a prime. These solutions are (2<i>F</i>, 2<i>F</i>) and (<i>F</i> + 1, <i>F</i>(<i>F</i> + 1)). We generalize an upper bound for the sum of solution terms from <i>n</i> = 3 established by Crilly and Fletcher in 2015 to any <i>n</i> to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mi>F</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>F</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and determine a lower bound to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mroot><mrow><msup><mi>n</mi><mi>n</mi></msup><mi>F</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mroot></mrow></semantics></math></inline-formula>. Confining the solutions to <i>n</i>-tuples consisting of distinct terms, equations of the 4th degree with any <i>F</i> are solvable but equations of the 5th to 9th degree are not. An upper bound for the sum of terms of distinct-term solutions is conjectured to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mi>F</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>[</mo><mrow><mi>F</mi><mo>+</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>!</mo><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>/</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>!</mo></mrow></semantics></math></inline-formula>. The conjecture is supported by computation, which also indicates that the upper bound equals the largest sum of solution terms if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>=</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mi>k</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>!</mo><mo>−</mo><mn>1</mn><mrow><mo>,</mo><mo> </mo></mrow><mi>k</mi><mo>∈</mo><msup><mo>ℤ</mo><mo>+</mo></msup></mrow></semantics></math></inline-formula>. Computation provides further insights into the relationships between <i>F</i> and the sum of terms of distinct-term solutions.Petr KarlovskyMDPI AGarticleDiophantine equationparametric solutionLagrange multipliersums and productsupper boundlower boundMathematicsQA1-939ENMathematics, Vol 9, Iss 2779, p 2779 (2021)

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