The diophantine equation \(x^ 2=4q^ n-4q+1\) (Q1262885)
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scientific article; zbMATH DE number 4125472
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The diophantine equation \(x^ 2=4q^ n-4q+1\) |
scientific article; zbMATH DE number 4125472 |
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The diophantine equation \(x^ 2=4q^ n-4q+1\) (English)
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1989
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The author completely solves the title equation for any given odd prime q. His method depends on factorization in a quadratic number field, and relating the solutions to the occurrences of \(\pm 1\) as values in a certain linear binary recurrence. Surprisingly, neither p-adic methods nor Gel'fond-Baker machinary are needed. When \(q=2\) the equation reduces to the well-known Ramanujan equation \(x^ 2=2^ n-7\), solved by Nagell in 1948.
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higher degree diophantine equation
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unique factorization
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linear binary recurrence
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Ramanujan equation
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