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We prove a Bombieri-Vinogradov type theorem for the number of representations of an integer $N$ in the form $N=p_1^g+p_2^g+...+p_s^g$ with $p_1,p_2,...,p_s$ prime numbers such that $p_1\equiv l\left(\mathrm{mod}k\right)$, under suitable hypothesis on $s=s\left(g\right)$ for every integer $g\ge 2$.