On a conjecture of Erdős and Stewart (Q2701574)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On a conjecture of Erdős and Stewart |
scientific article |
Statements
On a conjecture of Erdős and Stewart (English)
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19 February 2001
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Erdős-Stewart Diophantine equation
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\(p\)-adic linear forms in two logarithms
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lower bounds
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Erdős and Stewart conjectured that the only solutions of the equation NEWLINE\[NEWLINEn!+1=p^a_k p^b_{k+1}NEWLINE\]NEWLINE in integers \(n\geq 1\), \(a\geq 0\), \(b\geq 0\) and \(p_{k-1}\leq n < p_k\) where \(p_k\) denotes the \(k\)th prime number are obtained when \(n \leq 5\). The author confirms the conjecture. The proof depends on the lower bounds for \(p\)-adic linear forms in logarithms due to Bugeaud and Laurent and also on computer calculations which are not long.
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