Prime numbers are often described with an example like the following: 'if you have n counters, and can't make a rectangle which has both sides longer than 1, n is prime'
I think it would be interesting to see what happens if we extend this to 'if you have n counters, you can't make a y dimensional 'cuboid' which has all sides longer than 1, n is prime(y)'
I.e. what we currently call prime numbers would be defined as prime(2)
and the first prime(3) numbers would be 2,3,4,5,6,7,9,10,11,13 etc.
Has anyone already looked at this? Or is there some trivial connection with 2-primes and y-primes that I'm missing?
In number theory, the length of a natural number $n$ with a factorization like this: $$n=p_1^{r_1}\cdot\ldots\cdot p_m^{r_m}$$ is $r_1+\ldots+r_m$.
With your definition, a natural number is prime$(y)$ if its length is $\le y$.