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Keywords:
Diophantine equation; exponential sum; prime
Summary:
Let $[\theta ]$ denote the integral part of the real number $\theta .$ We prove that for $1<c<\frac {25 908}{18 905}$, the Diophantine equation $$ [p_{1}^{c}]+[p_{2}^{c}]+[p_{3}^{c}]+[p_{4}^{c}]+[p_{5}^{c}]=N $$ is solvable in prime variables $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ such that $p_1=x^2+y^2+1$ with integers $x$ and $y$ for sufficiently large integer $N$, and we also establish the corresponding asymptotic formula. This result constitutes a refinement upon that of S. Dimitrov (2023).
References:
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