On a Diophantine equation with five prime variables

Let $[x]$ denote the integral part of the real number $x$, and $N$ be a sufficiently large integer. In this paper, it is proved that, for $1<c<\frac{4109054}{1999527}, c\not=2$, the Diophantine equation $N=[p_1^c]+[p_2^c]+[p_3^c]+[p_4^c]+[p_5^c]$ is solvable in prime variables $p_1,p_2,p_3,p_4,p_5$.