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My mathematical training taught me that the good reason for 1 not being considered prime is the fundamental theorem of arithmetic, which states that every number can be written as a product of primes in exactly one way.
When I was young I thought that 1 was prime. The guy who explained that it wasn’t just said “they just decided that it’s not prime, a lot of stuff works out better that way”. A good, succint “a wizard did it” explanation that satisified me (after all, it had worked out for i, j, k and so on, I didn't trip out over the name "imaginary" and it pretty immediately turned out to be useful) and over the years I found that it was true, that a lot of things really did work out better.
But reading Lamb’s essay, I see that a wizard really did do it, i.e. that it really was something that the math community actually decided rather than calculated out.
High five to the original Mr Taxicab, G. H. Hardy:
Caldwell and Xiong cite G. H. Hardy as the last major mathematician to consider 1 to be prime.