Supernatural number

Contents

Definition

A supernatural number (also called a surnatural number, generalized natural number, or Steinitz number) is defined as a formal product of the form:

$\prod_p p^{n_p}$

where $p$ runs over all the prime numbers, and each $n_p$ is either zero, or a natural number, or $\infty$.

If none of the $n_p$s is $\infty$ and all but finitely many of them are zero, then the supernatural number can be identified with the natural number obtained by viewing the formal product as an actual product in the integers. The unique factorization of natural numbers tells us that every natural number has a unique representation as a supernatural number where none of the $n_p$s is $\infty$ and all but finitely many of them are zero.

Operations

For two supernatural numbers

Suppose we have two supernatural numbers:

$\! m = \prod_p p^{m_p}$

$\! n = \prod_p p^{n_p}$

We can define the following operations:

Operation	Output
product of supernatural numbers	$mn = \prod_p p^{m_p + n_p}$. Here the $+$ denotes usual addition if both $m_p$ and $n_p$ are finite. If either is $\infty$, the sum is $\infty$.
lcm of supernatural numbers	$\operatorname{lcm}(m,n) = \prod_p p^{\max \{m_p,n_p\}}$. Here, the $\max$ denotes usual maximum if both $m_p$ and $n_p$ are finite. If either is $\infty$, the maximum is $\infty$.
gcd of supernatural numbers	$\operatorname{gcd}(m,n) = \prod_p p^{\min \{m_p,n_p\}}$. Here, the $\min$ denotes usual minimum if both $m_p$ and $n_p$ are finite. If either is $\infty$, the minimum is the other number. If both are $\infty$, the minimum is $\infty$.

When both inputs are natural numbers, the definitions of these operations agree with the usual definition for natural numbers.

For many supernatural numbers

The definitions are similar to those for two supernatural numbers. Note that unlike the case of the natural numbers, we can take the product, lcm, and gcd of infinitely many supernatural numbers and still obtain a supernatural number. In particular, if we start with infinitely many natural numbers and perform one or more of these operations, we may end up with a supernatural number that is not a natural number.