# What is the set of all positive integers that can make a cuboid? Is there anything special about it?

About prime numbers, I know a geometric way to think about them. Can a number of unit squares make a rectangle whose dimensions are at least 2x2? If so, it's not prime.

1 square cannot, 2 squares cannot, 3 squares cannot, 4 squares can (2x2)... 5 cannot, 6 can (2x3), etc.

So the set of numbers that cannot do this is {1, 2, 3, 5, 7, 11, 13, 17, 19, ...}. You can see it's all prime numbers, except for 1.

NOTE: the reason for minimum dimensions of 2x2 is because any number n can make a 1xn rectangle. Therefore each dimension must be greater than 1 or there's nothing special going on. That's why "1" ends up in the set.

My next idea is to expand this into 3 dimensions. Can a number of unit cubes make a cuboid whose dimensions are at least 2x2x2?

Obviously 1 to 7 cannot. 8 can (2x2x2). 9, 10, 11 cannot. 12 can (2x2x3). 13, 14, 15 cannot. 16 can (2x2x4). 17 cannot. 18 can (2x3x3).

(Again, the reason for minimum dimensions of 2x2x2 is because a 1x1xn cuboid can be made with any number n. And a 1xnxm cuboid can be made with any number of the regular composites, so it would not give us anything distinct. Therefore, each dimension must be greater than 1. That's why "1" ends up in the set again.)

I don't know what to call these. Maybe cuboid primes?

Set of cuboid primes is {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, ...}

Set of cuboid composites is {8, 12, 16, 18, 20, 24, 27, 28, 30, ...}

I don't see any obvious patterns.

So, has this set of numbers been studied before? Does it have an official name? I would like to read more about them. What relation do they have to prime numbers? Is there anything special about the set in and of itself?

One thing I note...the gaps between cuboid primes start out very slowly (1 to 7 have no gaps), whereas the normal primes, the gaps in normal primes quickly increase (only 1 to 3 have no gaps).

I also noticed the cuboid primes still have adjacent numbers (no gaps) all the way out to 25 and 26...maybe more. This could never happen with normal primes, because all even numbers are composite except 2. So now I wonder, is there an even-odd analogy in 3 dimensions, and could it help us?

BTW if anyone can think of a special geometric reason to exclude 1 from both definitions, let me know.

• In case others search for this (and with respect to the answer below) the other name I thought of was 'Volumetric Primes'. Jun 15 '18 at 1:08

At the end of the day you're just counting the number of prime factors. If $n$ has at least $k$ prime factors (counted with repetition, so $8$ has three prime factors), then we can build a $k$-dimensional prism with each side of length at least $2$.

For example, $120$ has $5$ prime factors: $2, 2, 2, 3, 5$. Now consider the following arrangements:

• $2$ dimensions: $2$ by $(2\cdot 2\cdot 3\cdot 5)$.

• $3$ dimensions: $2$ by $2$ by $(2\cdot 3\cdot 5)$.

• $4$ dimensions: $2$ by $2$ by $2$ by $(3\cdot 5)$.

• $5$ dimensions: $2$ by $2$ by $2$ by $3$ by $5$.

Incidentally, the number of prime factors of an integer $n$, counted with multiplicity, is denoted "$\Omega(n)$," and so we can in principle denote the set of natural numbers with exactly $n$ many prime factors with multiplicity by $\Omega^{-1}(\{n\})$ or (in a slight abuse of notation) $\Omega^{-1}(n)$. However, I don't think this is generally done; unlike the primes, my understanding is that we only occasionally care about any specific "level" $\Omega^{-1}(n)$, and so there's no real need for a special name or symbol in general.

Re: $1$ being a special case, we can make geometric arguments for it - e.g. a $3$ by $1$ by $16$ prism is in our context "essentially" $2$-dimensional, since there's only one second coordinate that can occur - but the real reasons ultimately are algebraic: all the useful theorems about primes would have to be annoyingly reworded if $1$ was included (e.g. think about the uniqueness of prime factorizations).