supernatural number

A supernatural number ω is a formal productPlanetmathPlanetmath


where p runs over all (rational) prime numbersMathworldPlanetmath, and the values np are each either natural numbersMathworldPlanetmath or the symbol .

We note first that by the fundamental theorem of arithmeticMathworldPlanetmath, we can view any natural number as a supernatural number. Supernatural numbers form a generalizationPlanetmathPlanetmath of natural numbers in two ways: First, by allowing the possibility of infinitely many prime factorsMathworldPlanetmath, and second, by allowing any given prime to divide ω “infinitely often,” by taking that prime’s corresponding exponent to be the symbol .

We can extend the usual p-adic to these supernatural numbers by defining, for ω as above, vp(ω)=np for each p. We can then extend the notion of divisibility to supernatural numbers by declaring ω1ω2 if vp(ω1)vp(ω2) for all p (where, by definition, the symbol is considered greater than any natural number). Finally, we can also generalize the notion of the least common multipleMathworldPlanetmath (lcm) and greatest common divisorMathworldPlanetmathPlanetmath (gcd) for (arbitrarily many) supernatural numbers, by defining

lcm({ωi}) =ppsup(vp(ωi))
gcd({ωi}) =ppinf(vp(ωi))

Note that the supernatural version of the definitions of divisibility, lcm, and 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 subgroupsMathworldPlanetmathPlanetmath, in which case many of the theoremsMathworldPlanetmath from finite groupMathworldPlanetmath theory carry over verbatim.


  • 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