All the solutions of the equation \(\sum ^{11}_{i=1} \frac{1}{x_i}= 1\) in distinct integers of the form \(x_i \in 3^{\alpha} 5^{\beta} 7^{\gamma}\) (Q941363)
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scientific article; zbMATH DE number 5321300
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | All the solutions of the equation \(\sum ^{11}_{i=1} \frac{1}{x_i}= 1\) in distinct integers of the form \(x_i \in 3^{\alpha} 5^{\beta} 7^{\gamma}\) |
scientific article; zbMATH DE number 5321300 |
Statements
All the solutions of the equation \(\sum ^{11}_{i=1} \frac{1}{x_i}= 1\) in distinct integers of the form \(x_i \in 3^{\alpha} 5^{\beta} 7^{\gamma}\) (English)
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4 September 2008
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The author determines all the solutions \(x_1<x_2<\dots<x_{11}\) of the equation \( 1 = \sum_{i=1}^{11}\) in distinct integers \(x_i \in \{3^\alpha \cdot 5^\beta \cdot 7^\gamma;\; \alpha \geq 0,\, \beta \geq 0,\,\gamma \geq 0\}\). This equation has exactly 17 such solutions, all of which are given; the smallest value for \(x_{11}\) is \(x_{11}=135\).
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representation of integers
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sum of Egyptian fractions
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explicit solution of a special Diophantine equation in integers of the form \(3^\alpha 5^\beta 7^\gamma\)
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