Remington says they were serial numbered, but they don't say where on the gun.
My only Remington .22 is a 513, numbered on the bottom of the barrel just ahead of the foreend tip.
Targetmaster – Single Shot Bolt Action
Introduction Year: 1936
Year Discontinued: 1939
Total Production: Approximately: 307,000
Designer/Inventor: C.C. Loomis
Action Type: Single Shot Bolt Action
Caliber/Gauge: .22 Short, .22 Long, .22 Long Rifle
Serial Number Blocks: Starting: 00001
Ending: 307000
Grades Offered: Model 41 Targetmaster, 41A, 41PS, 41AS
Variations: Model 41A Smooth bore, Model 41SB