A ternary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$
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arXiv
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866916603571994624 |
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| author | Dimitrov, S. I. |
| author_facet | Dimitrov, S. I. |
| contents | In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $1<c<\frac{427}{400}$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$, such that $p_1=x^2 + y^2 +1$. For this purpose we establish a new Bombieri -- Vinogradov type result for exponential sums over primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_03967 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A ternary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$ Dimitrov, S. I. Number Theory In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $1<c<\frac{427}{400}$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$, such that $p_1=x^2 + y^2 +1$. For this purpose we establish a new Bombieri -- Vinogradov type result for exponential sums over primes. |
| title | A ternary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2011.03967 |