On the exceptional set for the sum of a prime and the \(k\)-th power of a prime (Q2714345)
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scientific article; zbMATH DE number 1604252
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the exceptional set for the sum of a prime and the \(k\)-th power of a prime |
scientific article; zbMATH DE number 1604252 |
Statements
13 June 2001
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sum of a prime and \(k\)-th power of a prime
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circle method
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exceptional set
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On the exceptional set for the sum of a prime and the \(k\)-th power of a prime (English)
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The author proves that ``almost all'' integers which feasibly could be expressed as the sum of a prime and the \(k\)th power of a prime, actually can be so expressed. By ``almost all'' the author gives the surprisingly strong upper bound \(O_k(x^{1-\delta_k})\) for the number of exceptional \(n\) up to \(x\). The proof is modelled on \textit{H. L. Montgomery} and \textit{R. C. Vaughan}'s application of the circle method [Acta Arith. 27, 253-370 (1975; Zbl 0301.10043)] to the Goldbach problem (the case \(k=1\) here), though extensively modified, and also influenced by several papers that have appeared in the meantime.
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