The equations \((x+1) \cdots (x+k) = (y+1) \cdots (y+m),\quad m=5,6\) (Q1815087)
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scientific article; zbMATH DE number 941340
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The equations \((x+1) \cdots (x+k) = (y+1) \cdots (y+m),\quad m=5,6\) |
scientific article; zbMATH DE number 941340 |
Statements
The equations \((x+1) \cdots (x+k) = (y+1) \cdots (y+m),\quad m=5,6\) (English)
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9 December 1996
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The title equation in integers \(x\geq 0\), \(y\geq 0\) and \(k\geq 2\), with fixed \(m\geq 2\), was completely solved for \(m=2,3,4\) by \textit{N. Saradha} and \textit{T. N. Shorey} in a number of papers [Proc. Indian Acad. Sci., Math. Sci. 100, 107-132 (1990; Zbl 0716.11017); Indag. Math., New. Ser. 2, 489-510 (1991; Zbl 0757.11010); ibid. 3, 79-90 (1992; Zbl 0757.11011)]. In the present paper, the authors give explicit upper bounds for \(k\) and \(y\) in terms of \(m\), in case \(5\leq m\leq 20\). These are derived from a more general result. For \(m=5,6\), they are able to sharpen their results and show that no solutions exist for these two values for \(m\).
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exponential equations
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upper bounds
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