That there exists no projective plane of order 6 is a consequence of the Bruck-Ryser theorem. The values up to n=7 may be verified by hand. The uniqueness of the projective plane of order 8 was proved by using a computer.
The four projective planes of order 9 were known for quite some time, but the proof that there doesn't exist another plane of this order, requires a computer.
Finally, the non-existence of the order n=10 needs pretty massive computer usage.