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Keywords:
Diophantine equation; prime; exponential sum; asymptotic formula
Diophantine equation; prime; exponential sum; asymptotic formula
Summary:
Let $[{\cdot }]$ be the floor function. In this paper, we prove by asymptotic formula that when $1<c<\frac {3441}{2539}$, then every sufficiently large positive integer $N$ can be represented in the form $$ N=[p^c_1]+[p^c_2]+[p^c_3]+[p^c_4]+[p^c_5], $$ where $p_1$, $p_2$, $p_3$, $p_4$, $p_5$ are primes such that $p_1=x^2 + y^2 +1$.
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