It is likely that your watch is from 1959, which is about right for the model. As has been discussed before, Rolex started over on their numbering system when they reached the 1 millionth oyster case, duplicating older numbers. You have to be a slight detective to figure these out but you have done well. As you noted the GMT was not available in 1945 so you knew that couldn't be correct. When you see a watch that is obviously 50's with a serial number indicating 30's, as with some of the first of these made after 1,000,000 you have to use ituition and dig further. The standard models are generally pretty obvious because of the hand and dial stylings. Also we know that many of the reference numbers simply were not made back then as is the situation here. Automatics were not made before 1934 so you won't find a 1929 perpetual. You won't find any calendar models made before 1945 or so. If someone is selling a datejust from the early forties you know that they have misinterpreted the serial number...This is rampant on the auction sites!!!! 1938 Oysterdate Precision NR....doesn't exist.

Inside your case back will likely be a Roman numeral I-IV and then a two digit Arabic number indicating the year of manufacture ie., 59. The Roman numeral indicates the quarter of the year it was manufactured in. The serial number can be estimated fairly close, by adding a 1 in front and the appropriate amount of zeros to make a full 7 digit number. In your case just the one will suffice making it 1,397,***. It has been said that the new cases started with 100,000 but there have been many cases that have shown up with 5 digit numbers so we know that there is some discrepancy with this theory. This information comes to me from a certified Rolex watchmaker.

Hope it helps out. Nice watch by the way...I wish I could afford it. What are you asking?