There can easily be confusion about the "-reme" description of the ancient shipping.

While bireme and trireme ships had two and three banks of oars, surprisingly enough quadriremes and quinquiremes did not have four and five banks of oars.

The usage was confused, but what seems to be the case is that the "tri-" in some cases, and the "quadri-" and "quinqui-" roots refer not to the number of banks of oars but to the number of rowers per oar. A greater number of oar-banks than three is mechanically impracticable. The angle of the fourth and fifth banks, if there were such, would mean that those oars would have to be way too long to be effective, and there would be real problems keeping the sets of oars from interfering with each other.

Dave Oesterreich