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A quinary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$
Published 29 Jun 2021 in math.NT | (2107.04028v3)
Abstract: In this paper we show that, for any fixed $1<c<\frac{5363}{3900}$, every sufficiently large positive number $N$ and a small constant $\varepsilon\>0$, the diophantine inequality \begin{equation*} |p_1c+p_2c+p_3c+p_4c+p_5c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4,\,p_5$, such that $p_1=x2 + y2 +1$.
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