Re: As Stakes Increase, Prime-Number Theory Moves Closer to Proof
- From: Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Mon, 11 Apr 2005 11:13:59 +1000
In article <1112979395.579443.86440@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Pubkeybreaker" <Robert_silverman@xxxxxxxxxxxx> wrote:
> Note that this is a popular press article. I would not expect it to
> use rigorous terminology.
>
> While it is quite clear that as written, the statement(s) are
> incorrect, there is a correct sense in which the primes ARE more
> numerous than the squares: by their arithmetic DENSITY in the
> integers. While both have density 0,
>
> lim n-->oo ( #primes < n / #squares < n) => oo.
>
> This is the concept that most people embrace, as opposed to
> cardinality of sets. (and perhaps it is the more ''inuitive')
>
> By this measure, the density of integers divisible by 2 (1/2) IS
> greater than the density of integers divisible by 9 (1/9).
>
> What non-mathematicians view as the concept of 'more numerous' means
> 'more numerous in sufficiently large finite subsets' because they
> can't cope with infinite set cardinalities.
>
> But of course they confuse 'density' with 'cardinality'.
To put what Bob has said another way:
I think what's really going on here is neither cardinality nor
density but rate of growth of the counting function. Look at the
examples cited in the article:
The number of numbers divisible by 2 is infinite, and so is the
number divisible by 9. But the first infinity is bigger. There also
is an infinite number of squares (4, 9, 16 ...) and cubes (8, 27,
64 ...), but more primes than either.
Density isn't enough - squares & primes both have density zero.
Rephrase it as the counting function for divisible by 2 is x / 2,
for divisible by 9 is x / 9, for squares is square root of x, for
cubes is cube root of x, for primes is (asymptotic to) x / log x,
and all the talk about bigger infinities is really about faster
growing functions. In a sense that can be made mathematically
rigorous, x / 2 ## x / 9 ## x / log x ## sqrt(x) ## cuberoot(x),
where ## is a symbol I just invented for "grows faster than."
It was not a great piece of math popularization, but with a bit
of good will you can work out what the author really meant.
--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.
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