Extending the primes

I had an idea and I'd like to find out whether it has a name or has been studied before.

Imagine the natural numbers and the operations of addition and multiplication, but with the following restriction: multiplication can only be carried out $d$ times, and only with prime numbers. Any number that usually has more than $d$ prime factors now becomes 'prime' in the new system. So, if $d=2$ say, then $8$ becomes a new 'prime' because the usual prime decomposition of $8= 2\times2\times2$ takes 'too many' multiplications; similarly, if $d=3$ then $16=2\times2\times2\times2$ becomes 'prime'.

Has this kind of thing ever been studied?

Edit: I've been asked to clarify this, so here's two other ways of describing the same object.

Geometrically. In a $d$-dimensional Euclidean space, let $\mathscr{D}$ be the smallest set of natural numbers such that you can represent every natural number as the volume of a $d$-dimensional box with edge lengths that are members of $\mathscr{D}$. $\mathscr{D}$ always contains the primes, and if $d<\infty$ it also contains other numbers. Perhaps it'd be interesting to understand those other numbers.

Algorithmically. First, put all the prime numbers in a set $\mathscr{D}$. Then, put every number $n$ with $\Omega(n)>d$ in $\mathscr{D}$ (where $\Omega(n)$ is the number of indistinct prime divisors of $n$). Finally, go through each of the numbers $n$ for which $\Omega(n)>d$ (i.e. the ones which were just added to $\mathscr{D}$) in increasing order, and for each such number $n$ that has not yet been removed from $\mathscr{D}$, remove from $\mathscr{D}$ every number $m$ that is product of $n$ and $d-1$ (possibly identical) numbers $k_1,k_2,...,k_{d-1}$ such that each $k_i\in\mathscr{D}$ and each $k_i \le n$. We can call the remaining numbers in $\mathscr{D}$ '$d$-prime'.

• But $16 = 8\times 2$ which only has two factors. – Tobias Kildetoft Jul 30 '13 at 10:19
• So now you want to have elements that are not allowed to be multiplied by anything else? – Tobias Kildetoft Jul 30 '13 at 10:25
• If $d=\infty$ this becomes $\mathbb{N}$; I'm interested in understanding the patterns of 'primes' created by restricting the number of indistinct prime factors a number can have. Probably I should have phrased it in a more number-theoretic way. – Marius Kempe Jul 30 '13 at 10:32
• But if not all numbers are allowed to be multiplied, then even speaking of primes does not really make sense. – Tobias Kildetoft Jul 30 '13 at 10:32
• I guess that Tobias (and I) is pressing you to define your new concept, call it prime*, rigorously. You shouldn't just call it "prime", if it also depends on the usual concept of prime, because otherwise it is impossible to tell which meaning of "prime" is meant at each instance of the word. – Jyrki Lahtonen Jul 30 '13 at 12:24

First, as Jyrki suggested, let's find some other term than prime. Maybe $d$-Mariusian is appropriate. We define the set $$\mathscr D_d:=\{n\in\mathbb N\mid n \text{ is d-Mariusian}\}$$ Now, forbidding multiplication is somehow difficult, so let's find a more formal definition for Mariusian numbers:

Definition: Let $n\in\mathbb N$. $n$ is called "$d$-Mariusian", if $n\neq 1$ and either $n$ is a prime or $$n=\prod_{i=1}^km_i \;\text{ for }\; m_1,\dots,m_k\in\mathscr D_d \text{ implies } k>d \text{ or } k=1$$

I hope, this agrees with your concept, otherwise skip the following and let me know.

Note, that $\mathscr D_1=\mathbb N\setminus\{1\}$ and $\mathscr D_{\infty}=\mathbb P$. Now, let's study $2$-Mariusianity.

If $n$ is prime, we have $n\in \mathscr D_2$. If $n=pq$ for primes $p,q$, then $pq\not\in\mathscr D_2$.

Let $n=pqr$ for primes $p,q,r$. Assume $n$ is not $2$-Mariusian, then it is the product of two $2$-Mariusian integers. There are only two (kinds of) ways to write $pqr$ as product of two integers:

• Seperate one prime, i.e: $n=p(qr)=q(pr)=r(pq)$, but then you have a non $2$-Mariusian factor.
• $n=(pqr)\cdot 1$, but $1\not\in\mathscr D_2$ (You can see here, that it is important to the well-definedness, that we defined $1$ to be not-Mariusian)

So $n=pqr$ is $2$-Mariusian. You can go on and see, that $$n=\prod_{i=1}^kp_i^{e_i}\in\mathscr D_2 \Leftrightarrow 2\nmid \sum_{i=1}^ke_i$$ Now you can try to give a characterisation for $d$-Mariusianity for arbtirary $d$'s.

Not an answer, but an attempt to formalize the OP's definition:

Let $M_d(S)$ be the set of all products of up to $d$ elements of $S$.

Define the sets $P_i$ and $C_i$ recursively as follows:

• $P_0$ are the prime numbers;
• $C_i = M_d(P_i) \setminus P_i$;
• $P_{i+1} = M_{d+1}(P_i) \setminus C_i$.

$P_i \subset P_{i+1}$, so we can say that a number is prime* if it is an element of $P_\infty$, i.e. an element of $P_i$ for some $i$.

Taking $d=2$ and looking only at powers of 2, we have that

• $P_0 = \{2\}$
• $C_0 = \{4\}$
• $P_1 = \{2,8\}$
• $C_1 = \{4,16,64\}$
• $P_2 = \{2,8,32,128,512\}$

and so forth. Marius, is this what you have in mind?