ON AN EQUATION INVOLVING FRACTIONAL POWERS WITH PRIME NUMBERS OF A SPECIAL TYPE

We consider the equation $[p^c_1]+[p^c_2]+[p^c_3] = N$, where $N$ is a sufficiently large integer, and $[t]$ denotes the integer part of $t$. We prove that if $1 < c < \frac{17}{16}$, then it has a solution in prime numbers $p_1, p_2, p_3$ such that each of the numbers $p_1 + 2, p_2 + 2, p_3 + 2$ has at most $\Big[\frac{95}{17−16c}\Big]$ prime factors, counted with their multiplicities.