In this paper, we show that a small minimal k-blocking set in PG(n, q), q = p, h ≥ 1, p prime, p ≥ 7, intersecting every (n−k)-space in 1 (mod q) points, is linear. As a corollary, this result shows that all small minimal k-blocking sets in PG(n, p), p prime, p ≥ 7, are Fp-linear, proving the linearity conjecture (see [7]) in the case PG(n, p), p prime, p ≥ 7.