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>...
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
|
| Materias: | |
| Acceso en línea: | https://doaj.org/article/84716101e415455d887f20b0defa87f3 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| Sumario: | 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. |
|---|