Let [θ] denote the integral part of the real number θ. We prove that for \(1<c<{25\,908 \over 18\,905}\) , the Diophantine equation \([p_{1}^{c}]+[p_{2}^{c}]+[p_{3}^{c}]+[p_{4}^{c}]+[p_{5}^{c}]=N\) is solvable in prime variables p1, p2, p3, p4, p5 such that p1 = x2 + y2 + 1 with integers x and y for sufficiently large integer N, and we also establish the corresponding asymptotic formula. This result constitutes a refinement upon that of S. Dimitrov (2023).
