On integers \(2(p+ia)\) not of the form \(a^k+\varphi (m)\) (Q519982)
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scientific article; zbMATH DE number 6699213
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On integers \(2(p+ia)\) not of the form \(a^k+\varphi (m)\) |
scientific article; zbMATH DE number 6699213 |
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On integers \(2(p+ia)\) not of the form \(a^k+\varphi (m)\) (English)
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31 March 2017
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By extending a result of \textit{F. Luca} and \textit{V. J. Mejía Huguet} [Bol. Soc. Mat. Mex., III. Ser. 15, No. 1, 11--21 (2009; Zbl 1241.11003)] the author proves the following: Theorem. Let \(a\geq 2\) be an even integer and let \(L\geq 1\) be an integer. Then for a sufficiently large \(x\), the number of primes \(p\leq x\) such that \(2(p + a), \ldots, 2(p+La)\) can not be expressed as \(a^k +\varphi(m)\) is at least \(C(a,L) \frac{x}{\log x}\), where \(k\) and \(m\) are positive integers, \(\varphi(m)\) is Euler's totient function and the constant \(C(a,L)\) depends on \(a\) and \(L\).
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power of \(a\)
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the Euler totient function
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prime
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Selberg's sieve method
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0.7436293959617615
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0.7391752004623413
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