anders at aleph.se
Thu Feb 7 23:24:03 UTC 2013
On 07/02/2013 22:19, David Lubkin wrote:
> How significant are prime numbers outside of number theory?
> I remember them popping up occasionally as a device in crafting
> a proof by contradiction, but that's about it.
Pretty significant. In group theory subgroups have sizes that are
factors of the size of the larger group, so prime-sized groups are
simple groups. This means that primes creep in everywhere group theory
applies (more or less everywhere), like in symmetries. Essentially any
part of discrete mathematics will occasionally deal with primes since it
involves counting and chunking stuff into similar-sized sets. And then
there are the periodical cicadas, who spend a prime number of years
underground before emerging, likely because it makes it hard for
predators to synchronize their population oscillations to them..
> Versus numbers that are of broad mathematical (and practical)
> significance, like -1, 0, 1, 2, i, e, pi, and phi.
You are comparing apples or oranges (or rather, apples and Rosaceae) -
these are individual numbers, while primes typically occur as a type of
numbers. You should compare primes to odd numbers, fractions or
irrational numbers instead. There I think they hold their own quite nicely.
Of the above list of numbers I think phi is overrated. Sure, there are
many pretty identities with it, but most are not terribly *deep*. The
just follow straightforwardly from it being the solution of x^2-x-1=0.
The only IMHO truly "deep" aspect of phi is that it is the winding
number of the last invariant torus in the KAM theorem, but even that is
a consequence of it being the least approximable irrational number,
itself a consequence of its very simple continued fraction (which
follows from the equation). And don't get me started about people going
starry eyed over its golden and magical properties.
Give me Euler's gamma constant any day! (I strongly recommend Havil's
book on gamma)
Future of Humanity Institute
Oxford Martin School
Faculty of Philosophy
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