The simple classical groups of dimension less than 6 which are (2,3)-generated

In this paper, we determine the finite classical simple groups of dimension n = 3, 5 which are (2,3)-generated (the cases n = 2, 4 are known). If n = 3, they are PSL_3(q), q ≠ 4, and PSU_3(q^2), q^2 ≠ 9, 25. If n = 5 they are PSL_5(q), for all q, and PSU_5(q^2), q^2 ≥ 9. Also, the soluble group PSU_3(4) is not (2,3)-generated. We give explicit (2,3)-generators of the linear preimages, in the special linear groups, of the (2,3)-generated simple groups.