supernatural number
A supernatural number $\omega $ is a formal product^{}
$\omega ={\displaystyle \prod _{p}}{p}^{{n}_{p}},$ |
where $p$ runs over all (rational) prime numbers^{}, and the values ${n}_{p}$ are each either natural numbers^{} or the symbol $\mathrm{\infty}$.
We note first that by the fundamental theorem of arithmetic^{}, we can view any natural number as a supernatural number. Supernatural numbers form a generalization^{} of natural numbers in two ways: First, by allowing the possibility of infinitely many prime factors^{}, and second, by allowing any given prime to divide $\omega $ “infinitely often,” by taking that prime’s corresponding exponent to be the symbol $\mathrm{\infty}$.
We can extend the usual $p$-adic to these supernatural numbers by defining, for $\omega $ as above, ${v}_{p}(\omega )={n}_{p}$ for each $p$. We can then extend the notion of divisibility to supernatural numbers by declaring ${\omega}_{1}\mid {\omega}_{2}$ if ${v}_{p}({\omega}_{1})\le {v}_{p}({\omega}_{2})$ for all $p$ (where, by definition, the symbol $\mathrm{\infty}$ is considered greater than any natural number). Finally, we can also generalize the notion of the least common multiple^{} (lcm) and greatest common divisor^{} (gcd) for (arbitrarily many) supernatural numbers, by defining
$\mathrm{lcm}(\{{\omega}_{i}\})$ | $={\displaystyle \prod _{p}}{p}^{sup({v}_{p}({\omega}_{i}))}$ | ||
$\mathrm{gcd}(\{{\omega}_{i}\})$ | $={\displaystyle \prod _{p}}{p}^{inf({v}_{p}({\omega}_{i}))}$ |
Note that the supernatural version of the definitions of divisibility, $\mathrm{lcm}$, and $\mathrm{gcd}$ carry over exactly from their corresponding notions for natural numbers, though we can now take the gcd or lcm of infinitely many natural numbers (to get a supernatural number).
Supernatural numbers are used to define orders and indices of profinite groups and subgroups^{}, in which case many of the theorems^{} from finite group^{} theory carry over verbatim.
References
- 1 [Ram] Ramakrishnan, Dinikara and Valenza, Robert. Fourier Analysis on Number Fields. Graduate Texts in Mathematics, volume 186. Springer-Verlag, New York, NY. 1989.
- 2 [Ser] Serre, J.-P. (Ion, P., translator) Galois Cohomology. Springer, New York, NY. 1997
Title | supernatural number |
---|---|
Canonical name | SupernaturalNumber |
Date of creation | 2013-03-22 15:23:36 |
Last modified on | 2013-03-22 15:23:36 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 7 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 20E18 |
Defines | gcd of supernatural numbers |
Defines | greatest commond divisor of supernatural numbers |
Defines | lcm of supernatural numbers |
Defines | least common multiple of supernatural numbers |