Some applications of the \(abc\)-conjecture to the Diophantine equation \(qy^{m}=f(x)\) (Q2901984)
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scientific article; zbMATH DE number 6062710
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Some applications of the \(abc\)-conjecture to the Diophantine equation \(qy^{m}=f(x)\) |
scientific article; zbMATH DE number 6062710 |
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1 August 2012
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abc-conjecture
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Diophantine equation
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Some applications of the \(abc\)-conjecture to the Diophantine equation \(qy^{m}=f(x)\) (English)
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The author proves that the abc-conjecture implies the following assertion: if \(f\) is a completely reducible polynomial over the rationals of degree \(>2\), then the equation \(qy^m=f(x)\) has only trivial rational solutions in \(x\) and \(y\), for all but finitely many primes \(q\). Some other related statement are also proved. The results extend a theorem of Granville.
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