@futurebird one more way to thing about it: imagine a half-line of points / vectors with non-negative integer coordinates. There is a zero, and there is a "smallest" (cannot be represented as sum of two others) vector, 1.
Now imagine a quarter-plane, there will be two "smallest" vectors besides zero. They're interesting because we can represent any other vector in our quarter-plane as a sum of these "smallest" vectors (and not just sum but an unique sum). Of course we're interested in smallest non-zero vectors, otherwise zero vector would be the only smallest one. What we're interested in are "generating" vectors, those that define a shape of that quarter-plane and its content, and zero vector doesn't define anything.
We can then do the same exercise with 1/8th of 3-dimensional space, etc.
Now extend this to the space with countably many dimensions (and vectors with finite number of non-zero coordinates). And define the mapping between this space and positive integers: vector with a_i coordinate at ith place is converted to the product of ith prime numbers to the a_ith degree. Then vector addition turns into integer multiplication, "smallest" vectors turn into their respective primes, and origin / zero vector is converted to 1.
1 is a prime in the same sense as zero is the smallest vectors, but this doesn't get is anywhere, we're interested in smallest non-zero vectors, those that generate everything else.
futurebird