On a Diophantine equation involving primes

Abstract

Let $$[\theta ]$$ denote the integral part of the real number $$\theta $$ . In this paper it is proved that for $$1<c< \frac{137}{119}$$ , the Diophantine equation $$ [p^c_1] + [p^c_2] + [p^c_3] = N$$ is solvable in prime variables $$p_1, p_2, p_3$$ for sufficiently large integer N. The range $$1<c< \frac{137}{119}$$ constitutes an extension of $$1< c < \frac{258}{235}$$ due to Zhai and Cao.