That's a very late gun probably 1933, see related links for date code stamp.
You will have to contact Remington.
Your Remington model 1903-A3 was made in the year 1942.It was chambered in the caliber 30-06.
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The Remington model 4 rolling block rifle was made between 1890-1933.If you have a model 4S military rifle,then these were made by Remington from 1913-1923.your rifle has either a 22.5in barrel,or a 24in barrel.These were chambered in .22,.25 Stevens,or 32 caliber.
Impossible to tell unless you have the factory records.
List the caliber and then someone will try to help you.
Model 581.
Are you sure of the model number?
$500 up
SN's were not required before 1968
1920
1918