Re: About random, primes and statistics
- From: tommy1729 <tommy1729@xxxxxxxxx>
- Date: Wed, 22 Aug 2007 08:44:10 EDT
quasi wrote:
A few corrections are needed.
I'll do a full repost, mostly pasted from my previous
posts, but also
incorporating the corrections.
Here's the latest version ...
I am going to try to define what it means for a
deterministic sequence
to be, in some sense, "random".
After defining the concept, I'll state a conjecture
which, I think,
captures the spirit of JSH's claims for the
randomness of the sequence
of primes, modulo m.
The concepts are intuitive, but not so easy to make
precise.
I'll give it a try.
Let N denote the set of natural numbers.
Let x = x_1, x_2, x_3, ... be a sequence with values
in a given set.
The values can be of any type -- no restriction.
Let Y = range(x) = {y | y = x_n for some n in N}.
Note -- range(x) is a set, not a sequence.
For each y in Y, n in N, define the n'th partial
relative frequency
(kind of like a partial sum) by
prf_n(y) = #{ x_i | 1<=i<=n and x_i=y} / n
where # denotes cardinality.
Call x "eventually identically distributed" (EID) if
the following two
conditions are satisfied:
(1) "all partial relative frequencies approach
limits" (possibly 0)
More precisely ...
For each y in Y, lim [n -> infinity] prf_n(y)
(y) exists
(2) "the sum of the (limiting) relative frequencies
is 1"
More precisely ...
Let rf(y) = lim [n -> infinity] prf_n(y)
Then sum [y in Y] rf(y) = 1
Next we define a uniform version ...
Let x be an EID sequence. Call x "eventually
uniformly distributed"
(EUD) if the following conditions hold
(1) {y | rf(y) > 0} is finite.
(2) If a = #{y | rf(y) > 0}, then rf(y) > 0 =>
rf(y) = 1/a.
In other words, all y values with positive relative
frequency have the
same relative frequency.
Next, we need to define vector partial relative
frequencies.
Let x be an EID sequence.
Fix a positive integer k.
Let V=Y^k be the set of k-tuples of elements of Y.
For each v in V, n in N, n>=k, define the n'th vector
partial relative
frequency of v by
vprf_n(v) = #{ ( x_i, x_(i-1), ..., (x_(i-k+1)) |
) | k<=i<=n
and ( x_i, x_(i-1), ...,
and ( x_i, x_(i-1), ..., (x_(i-k+1)) = v}
divided by
(n-k+1)
In other words, the vprf_n(v) measures the relative
frequency within
the partial sequence x_1, ..., x_n of k successive
terms of the
sequence x which match the vector v (relative to the
number of
subsequences of k successive terms).
Call x "eventually locally random" (ELR) if x is EID
and, for all
positive integers k and all v in Y^k, the following
conditions are
satisfied:
(1) lim [n -> infinity] vprf_n(v) exists
(2) Let vrf(v) = lim [n -> infinity] vprf_n(v)
Then vrf(v) = prod [1<=j<=k] rf(v_j)
Similarly, x is "eventually locally uniformly random"
(ELUR) is
defined in the same way as above except x is required
to be EUD, not
merely EID.
Based on these concepts, I have a number of tentative
conjectures, but
for now, I'll just state a revised version of JSH's
claim ...
Conjecture (JSH) [as interpreted by quasi] :
Let m>2 be an integer. Then the sequence of primes
modulo m is
eventually locally uniformly random.
quasi
how about this definition of (deterministic) randomness of a mod 3 ;
similar to a base 3 expansion of a normal number.
this is way shorter , however it contains the perhaps harder concept of a "normal" number.
note that your intepretation is equal to mine.
and i think it is the only possible sensible intepretation of JSH ...
however once agian the idea is not new.
Dirichlet showed that primes = x mod "any prime" are equidistributed for x.
that is the probability of having x is simply 1/phi("any prime") , which simplifies to 1/("any_prime"-1).
so Dirichlet showed that the probability of 1 mod 3 = probability of 2 mod 3.
also tommy1729 has consider this in his factoring tricks and posts about mod.
He has found structures in primes that are of polynomial type ( 1 mod m )
combining those (tommy1729 and Dirichlet)
we get a typical for primes kind of result:
"apparantly" somewhere between pseudorandom , determined and structure.
Despite that, results may still occur since Dirichlet nor tommy1729 disproof the existance of
"some deeper structure".
However as indicated Dirichlet proved randomness of some kind...
He proved the density is random with respect to random picking...
SO JSH IDEA is not new.
nor does he have results ; not for randomness ; not for structure.
however the idea is intresting...
greets
tommy1729
.
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