On the factorization of \(f(n)\) for \(f(x)\) in \(\mathbb Z[x]\) (Q2840301)
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scientific article; zbMATH DE number 6189009
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the factorization of \(f(n)\) for \(f(x)\) in \(\mathbb Z[x]\) |
scientific article; zbMATH DE number 6189009 |
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17 July 2013
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linear forms in logarithms
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polynomials
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greatest prime divisor
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0.9132191
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0.9041819
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0.90302825
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0.8958851
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On the factorization of \(f(n)\) for \(f(x)\) in \(\mathbb Z[x]\) (English)
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Let \(f\) be a polynomial in \(\mathbb Z[X]\) with at least two distinct roots. In 1921 Siegel proved that the greatest prime factor of \(f(n)\) tends to infinity, the proof being ineffective. Here the authors consider a finite set \(S\) of prime numbers, as usual they define \(|m|_S = \prod_{p \in S} |m|_p^{-1}\) for a non-zero integer \(m\), and they prove effective upper bounds for \(|f(n)|_S\). This extends previous results of Stewart, and Bennett-Filatesa-Trifonov.
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