Despite the vast difference in numbers, your two BSA shotguns were despatched from the factory at Small Heath, Birmingham, only two weeks apart!
No.37473 was despatched on on July 30, 1923 and the other on August 13, 1923.
The serial numbers of BSA shotguns bear no relationship to their date of despatch.
I own and use a matched pair with a difference of 275 between their serial numbers, even though both guns were finished and despatched on the same day, also in 1923.
The BSA shotgun was introduced in 1920 and, over the next 19 years, more than 48,000 were built in eight different models of 12-bore.
Some 40,000 were sold in the first five years of production.