As a collector/buyer of guns for around 45 years, I can give you a few examples.
1. I collect .22/32 HFT's commonly known as the Bekeart model. Gun 138226 the first serial number used in the first 1,000 or so guns produced did NOT ship to Bekeart. 138227 did. There were 6 shipments to Bekeart out of that first 1,000 or so gun run so having the complete serial number is the only way to know which guns shipped to Bekeart and in which dated shipment.
138XXX tells me nothing and neither does 13822X.
2. I own Texas Ranger Commerative TR555. Since acquiring that gun, I have tried to find TR5555. When I see one for sale without the serial number shown, I have to ask the seller for the complete serial number. Sometimes they respond and sometimes they don't.
3. I own the .22 Outdoorsman pictured in my Avatar above. The gun, according to Roy Jinks was part of a two gun shipment. I have the other serial number. When I see an early Outdoorsman for sale without serial number listed or shown, I have to ask the Seller for it to see if it is the second gun in that shipment.
4. Roy Jinks has given out specific serial numbers that have come up during his research that have some significance or shipped to someone or some agency deemed special or neat. When searching for guns online, without the full serial number, there is no way to know if the gun for sale fits one of those unique situations.
So there you have a few reasons why full serial numbers are of value to a collector. If you are a shooter looking for a model 19-3 then you probably don't care what the serial number is.
