It has often been supposed that these are parallel accounts of the same incident. To be sure, both involve a centurion, in Capernaum, a sick servant, and the statement of the centurion along with the Lord's reaction are very similar. But other details simply do not match. Evidently the Romans had an army base in Capernaum, with a centurion as commanding officer, who could be rotated. [Where do you suppose Peter sold most of his fish? And what language did he use?] Looking at the sequence of events in both Matthew and Luke, I would say that the incident recorded by Matthew happened first, and a number of months before the one recorded by Luke. Of course an incident like that would become part of the ‘folklore’ of the base. I assume that the centurions were different, but they certainly knew each other, so the second one knew every detail of the first incident. When his turn came, he used a different strategy to make his appeal (he was asking for a second favor), but then repeated the statement that had impressed Jesus so favorably. So, the first centurion left his house, but the second did not.