Re: JSH: Understanding prime gaps issue



On Jul 16, 10:00 am, David Bernier <david...@xxxxxxxxxxxx> wrote:
MichaelW wrote:
On Jul 15, 10:05 am, JSH<jst...@xxxxxxxxx>  wrote:
On Jul 13, 7:32 pm, MichaelW<ms...@xxxxxxxxxx>  wrote:

On Jul 14, 10:25 am, JSH<jst...@xxxxxxxxx>  wrote:

Even a novice math student SHOULD know that the world does not
supposedly have a prime gap equation as with one you could prove or
disprove the Twin Primes Conjecture as well as an infinity of
conjectures of that type.

So my claim of having the world's only prime gap equation is a
fascinating one.

Someone with basic math knowledge who thinks I'm full of crap would
just reject it outright as a false statement.

My own claim follows from my prime residue axiom, and is tantamount to
a claim of having proven the Twin Primes Conjecture.

Search in Google: prime gap equation

James Harris

James,

Your equation gives the *probable* number of pairs of a certain gap
between primes not the *actual* number. You would need the actual

That's in accord with the prime residue axiom.

number to prove the TPC. In fact your work is a restatement of the
first Hardy-Littlewood conjecture which is considered most likely
true, but unproven.

My work says that it is all about probability and the lack of a prime
preference.

That is HUGE statement.  If true it indicates there is nothing else to
do!

With my prime gap equation you have predictions for ANY prime gap, as
well as theory as to when a particular gap can possibly occur, along
when it is likely to occur, so you can explain that with twin primes
or other gaps.

That's data dude.  If, say, someone checked the first million prime
gaps, and my prime gap equation worked well--it is probability so it
wouldn't be exact--then that would be hard to explain as accidental.

But it also ends the need to fund any other research in the area.

I've noticed only you and "Mark Murray" were stupid enough to come out
and claim that a prime gap equation already exists while other posters
have danced around the issue.

So, if someone wishes to reply that a prime gap equation is
established they should cite a source, no need to give a link, I don't
chase links.  And they should explain why it isn't being used to
resolve all prime gap conjectures!

I'd actually like ANY other poster besides "MichaelW" or "Mark Murray"
to reply in this thread definitively that their position without any
shadow of a doubt is that a prime gap equation already is known.

Anyone?  Anyone?  Anyone?

James Harris

Just to clarify my position:

(1) The prime gap equation I am referring to was posted on JSH's blog
dated Aug 9, 2006. This is the one that comes up in a Google search.

(2) The equation gives an estimate of the number of pairs of primes
with a given gap, not the actual number. In the example given an
estimate in a particular range is 1.5 and the actual number is 1 which
is "expected". It is not clear why the answer of 2 is not expected.

(3) Since the equation gives an estimate instead of an exact value it
does not provide a solution for the Twin Prime Conjecture (or the more
general form:http://en.wikipedia.org/wiki/De_Polignac%27s_conjecture
). The claim does however appear to be made that for higher number
ranges the estimate approaches the correct number.

(4) The equation is based around the multiplication of the terms (p-2)/
(p-1) for all primes from 3 to p_j where p_j is the j'th prime. This
formula rapidly approachs c / ln(p_j) where c ~ 0.741.

(5) Realising (4) it is possible to show that the prime gap equation
and particular cases of the First Hardy-Littlewood conjecture (
http://mathworld.wolfram.com/k-TupleConjecture.html) are essentially
the same.

(6) The main difference between JSH's equation and the H-L conjecture
is that James leaves out a constant term. For this reason the JSH
equation always gives the wrong answer by a constant factor. This is
not obvious at the very small values that are used to demonstrate the
equation but a few regulars here have posted trials over large number
ranges that clearly show the constant factor is missing. I have never
understood why James does not see this as in fact validating his
equation.

Thomas Nicely has been counting twin primes very extensively.
He has a page giving, among other things,
counts of twin primes below K for a large range of quite large
values of K. In a second column, the Hardy-Littlewood estimate
of the form (number of pairs below K) ~= A int_{2...K) (dt/(log(t)^2)),
A being given explicitly in H-L conjecture;  in a third column,
delta_2(K) := H-L estimate up to K - (actual count up to K).

A sign that a conjecture with probabilistic "overtones" is
doing well is [sometimes] that [delta(K)]^2 is of the same order as
(actual count up to K).  A cursory examination of
Nicely's table shows that this seems about right.

Nicely has the distinction of having discovered the
so-called "FDIV bug" in the arithmetic of an
Intel chip; a large recall followed, and my impression
is that for a time, it must have been quite embarrassing
for Intel ...

Reference to Nicely's count of twin primes up to large K, plus
estimates of Brun's constant:

<http://www.trnicely.net/twins/t2_0000.htm> .

Example:  x = 10.
pi_2(10) = 2 because of the twin prime pairs: (3,5) and (5,7).
pi_2(30) = 5  because of the five twin primes pairs:
              (3,5), (5,7), (11,13), (17,19), (29, 31).

Thomas Nicely writes:

<< pi_2(x) = Number of twin-prime pairs (q, q+2) such that the smaller element q
<= x. >>  [Definition].

If x = 30, the prime 29<=30 so the twin prime pair (29, 31) = (29, 29+2) counts
in pi_2(30), even though 31 > x and x=30.

Tomas Oliveira e Silva (Portugal) reports:

x = 1600d15
pi2(x) = 1264267586627937

2*C_twin * li2(x) = 1264267596740960.579...

1600d15 is 1600. 10^15 = 1.6 * 10^18.

<< Ctwin is the twin primes constant (0.66016181584686957392...) >>.

<< li2(x) is the principal value of the integral of 1/log^2 between 0 and x >>

So li2(x) is very close to int_{t=2 ... x} (dt/log(t)^2).
For x = 1.6*10^18, PARI-gp finds:

1264267596740960.579 - 1264267586627937 = 10113023.579,

and 10113023.579^2 = 102273245909409.969 < 1264267586627937/12 .

==========

Consider for arbitrarily large x: q(x) := pi(x + log(x) ) - pi(x).
One would expect q(x) to be about 1 on average.

Furthermore, obviously as x->oo, log(x) -> oo.

So what is limsup_{x->oo} q(x) ?

For every prime p between x and x+log(x), the remainder upon dividing by
a prime p' < x must be non-zero.  This will tend to increase the chances
that a random number between x and x+log(x) , other than p, is divisible by p'.

So it's not clear to me how q(x) grows with x, or that q(x) is unbounded.

David Bernier



(7) Time to define terms: is a prime gap equation "known"? What I have
meant (Mark Murray can speak for himself) is that what James has
produced is known, following from (5).  What James has meant (I think)
is that his equation gives the actual answer and he correctly states
that previously there has been no known equation to correct predict
the number of prime pairs of a certain gap over a range of numbers. My
point is that his equation does not give the actual answer either.

(8) Combining (4) with the prime number theorem it is possible to
construct much simpler equations to estimate the number of prime pairs
of a certain gap in a range of numbers. Despite James' claims to the
contrary this extending of his reseach was not welcomed.

I welcome but do not expect any feedback.

Regards, Michael W.

David,

Thanks for an especially cool reply. It is good so see someone
discussing maths rather than personalities or search ranks on this
forum!

The link is terrific and sufficiently text-like to be quotable going
forward. The results provided can be used to verify (at a basic level
since only twins aka k=2 is being counted) any equation claiming to
give a probably result of twin prime distribution.

My back of the envelope maths would indicate that q(x) = pi(x+ln(x) -
pi(x) should approximate some constant (probably 1) as x->oo but I
have been burnt before and would want to crunch the numbers first.

James,

Are you open to starting a maths only discussion of prime gap counts?
There is a lot of interest and expertise here by the looks.

Regards, Michael W.
.



Relevant Pages

  • JSH: Prime Gaps, Schnapps or Cups
    ... [note this is clean Non-JSH Math. ... A prime gap of length is a run of consecutive composite numbers between ... Setting reduces to Cramer's conjecture for large, ... Comput. ...
    (sci.math)
  • Re: JSH: Prime Gaps, Schnapps or Cups
    ... The maximal prime gap  is the length of the largest prime gap that begins ... Setting  reduces to Cramer's conjecture for large, ... Math. ... Comput. ...
    (sci.math)
  • Re: JSH: Why of prime gaps
    ... the key to understanding the twin prime gap. ... would have been a second twin primes case, ... could the prime 13 decide that it doesn't like a particular residue ... when I look up your Prime Residue Axiom* I get ...
    (sci.math)
  • Re: JSH: Why of prime gaps
    ... the key to understanding the twin prime gap. ... would have been a second twin primes case, ... could the prime 13 decide that it doesn't like a particular residue ... when I look up your Prime Residue Axiom* I get ...
    (sci.math)
  • JSH: Why of prime gaps
    ... the key to understanding the twin prime gap. ... would have been a second twin primes case, ... bigger primes p_1, and you are getting all these residues modulo ... residue modulo ANY of those primes which will give you a prime gap. ...
    (sci.math)