Consecutive square-free values of the type x2 + y2 + z2 + k, x2 + y2 + z2 + k + 1

We show that for any given integer k there exist infinitely many consecutive square-free numbers of the type x2 + y2 + z2 + k, x2 + y2 + z2 + k + 1. We also establish an asymptotic formula for 1 ⩽ x, y, z ⩽ H such that x2 + y2 + z2 + k, x2 + y2 + z2 + k + 1 are square-free. The method we used in this paper is due to Tolev.