## my elementary method and Perry's /Goldbach's conjectures

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• The following is an elementary way to attack Goldbach s conjecture, the Twin Primes conjecture and many other problems in prime number theory. Let
Message 1 of 8 , Jul 3, 2003
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The following is an elementary way to attack Goldbach's conjecture,
the Twin Primes conjecture and many other problems in prime number
theory.

Let Sn={1,2,...,n}. We will be interested in studying the subsets of
the set Sn, their cardinalities, and their characteristic functions.
The following subsets of Sn will be useful.

Pn={p| p is an odd prime and 3<=p<= n-3}
|Pn| = Pi(n-3)-1 where Pi(n) is the prime counting function. Let p
(x) = {1 if x is in Pn , 0 otherwise That is p(x) is the
characteristic function of the set Pn.

Cn = {c| c is composite and 3<=c<=n-3}
|Cn|= n - |Pn|. Let c(x) ={1 if x is in Cn, 0 otherwise That is c(x)
is the characteristic function of the set Cn.

Tn={t| t is relatively prime to n and 3<=t<=n-3}
|Tn| = Phi(n)-2 where Phi(n) is Euler's totient function. Let t(x) =
{1 if x is in Tn, 0 otherwise

Def. Sentence & equivalence
A sentence is a finite products and sums of characterisic functions
of subsets of Sn. Two sentences are equivalent iff the have the same
truth tables.

Example 1.

The following is a sentence with its value. Let n be an integer then
c(x)*c(n-x)*t(x) = {1 if x and n-x are both composite and (n,x)=1 and
0 otherwise

Example 2.
Let n be an integer then
c(x) is equivalent to 1-p(x) for all x | 2<= x <=n

Def. Sum over a sentence
Let Q be a sentence then we associate with Q the value Q+ which is
defined as the sum over all x in Sn {Q(x)}.

Lemma 1
If Q and W are two equivalent sentences then Q+=W+.
Proof: Since Q(x) = W(x) for all x in Sn the result follows trivially.

From this theory we have the following applications

Goldbach's Problem

Let Gn = { p | n-p and p are in Pn and (n,p)=1 } that is the set of
all primes p such that n-p is prime and p is not equal to n-p.
Let g(x) = {1 if x is in Gn, 0 otherwise

Lemma 2
g(x) is equivalent to p(x)p(n-x)t(x) for all x such that 3<=x<=n-3

Proof:
We may prove the lemma by considering the truth tables associated
with the sentences g(x) and p(x)p(n-x)t(x).

Let Hn={c | c and n-c are composite and (n,c)=1}
Let h(x) = {1 if x is in Gn, 0 otherwise
This function is associated with Perry's conjecture which is
available at http://www.primepuzzles.net/conjectures/conj_033.htm
We will prove this conjecture below. This lemma will be useful.

Lemma 3
h(x) is equivalent to c(x)c(n-x)t(x) for all x in Sn

Proof:
Follows immediately by considering truth tables.

Theorem 1
|Hn| = |Gn| + Phi(n) - 2Pi(n) + 2Omega(n) - 2 where Omega(n) is the
number of prime divisors of n.

Proof:
h(x) is equivalent to c(x)c(n-x)t(x) for all x in Sn by lemma 3.
c(x) is equivalent to 1-p(x)
so c(x)c(n-x)t(x) is equivalent to
(1-p(x))(1-p(n-x))t(x)
which is equivalent to
Q = t(x) - p(x) - p(n-x) +p(x)p(n-x)t(x)
now we find Q+ by taking the sum over Q and we find
Q+ = Sum(t(x) - p(x)t(x) - p(n-x)t(x) + p(x)p(n-x)t(x) )
splitting up this sum and noting that
Sum(t(x)) = Phi(n) - 2
Sum(p(x)t(x)) = Sum(p(n-x)t(x)) = Pi(n-3)- Omega(n) for n even (if
you want a proof notify me)
and
Sum(p(x)p(n-x)t(x)) = |Gn|
the result now follows by adding these identities together.
#

For many examples of Theorem 1 see my previous post.

Theorem 2 (Perry's Conjecture)

Every number number greater than 210 can be written as the sum of two
relatively prime composites.

Proof: (sketch)

We know by Theorem 1 that
|Hn| >= Phi(n) - 2Pi(n)

The result follows immediately from inequalities for Phi(n) - 2Pi(n)
for n sufficiently large then verifying the remaining cases.
#

I would give a complete proof to Theorem 2 but I am away from the
university at this moment and do not have access to the online
journals. I will post a complete proof at a later time.

Other Theorems and Future Research

Let R(n) be the number of representations of a number as the sum of
two primes then

Theorem 3

R(n) =Sum(over primes x<= n-3) Pi(n-x+1)-Pi(n-x-1)
Proof:
we may note that Pi(n-x+1)-Pi(n-x-1) is equivalent to p(n-x) and
since x is prime the result follows.
#

Theorem 4

Sum(even x | 6<=x<=n)R(x) = Sum(primes p <=n-3)(Pi(n-p+1)

The proof is a bit long and I will omit it here. There is a simple
proof base on another method which I will post at a later time.

I used this method mentioned here to study the sentence p(x)p(x+2) in
its equivalent forms and the method has shown some promise. A proof
of Goldbach's conjecture also seems possible by this method but my
limited time does not allow me to work with on the conjecture.

I would also like to study Dirichlet products of characteristic
functions and formal power series of such functions. The methods
outlined in Tom Apostol's Introduction to analytic number theory for
partial summation also seem to be helpful to establish some importan
inequalities. I am not ready to post any results on this front
though.

• Could we pass a law? Anyone submitting proofs must typeset them nicely Please? It s a dream anyway... Andy
Message 2 of 8 , Jul 3, 2003
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Could we pass a law?

'Anyone submitting proofs must typeset them nicely'

Andy
• ... I apologize. I will make a pdf file and a mathematica program with the paper and post a link to them. I would also like to state that I do not claim to
Message 3 of 8 , Jul 3, 2003
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<umistphd2003@y...> wrote:
>
> Could we pass a law?
>
> 'Anyone submitting proofs must typeset them nicely'
>
> Please? It's a dream anyway...
>
> Andy

I apologize. I will make a pdf file and a mathematica program with
the paper and post a link to them. I would also like to state that I
do not claim to prove Goldbach's conjecture.
• Hey, its ok, I was only half serious. It s difficult to set it out in plain text anyway! Andy ... From: antonioveloz2 To:
Message 4 of 8 , Jul 3, 2003
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Hey, its ok, I was only half serious. It's difficult to set it out in plain
text anyway!

Andy

----- Original Message -----
From: "antonioveloz2" <antonioveloz2@...>
Sent: Thursday, July 03, 2003 10:52 PM
Subject: [PrimeNumbers] Re: my elementary method and Perry's /Goldbach's
conjectures

> --- In primenumbers@yahoogroups.com, "Andy Swallow"
> <umistphd2003@y...> wrote:
> >
> > Could we pass a law?
> >
> > 'Anyone submitting proofs must typeset them nicely'
> >
> > Please? It's a dream anyway...
> >
> > Andy
>
> I apologize. I will make a pdf file and a mathematica program with
> the paper and post a link to them. I would also like to state that I
> do not claim to prove Goldbach's conjecture.
>
>
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
• Your proof is a bit difficult to follow, so I can t comment much. But it seems extremely suspicious that the Omega function should appear like that in theorem
Message 5 of 8 , Jul 3, 2003
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Your proof is a bit difficult to follow, so I can't comment much. But it
seems extremely suspicious that the Omega function should appear like that
in theorem 1.

But surely someone's proved Perry's "conjecture" already...?

Surely?!?

Andy
• ... But it ... like that
Message 6 of 8 , Jul 3, 2003
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<umistphd2003@y...> wrote:
>
> Your proof is a bit difficult to follow, so I can't comment much.
But it
> seems extremely suspicious that the Omega function should appear
like that
> in theorem 1.
>
> But surely someone's proved Perry's "conjecture" already...?
>
> Surely?!?
>
> Andy
• ... But it ... like that ... Sorry about the last post. I actually meant to print the following. We are considering the sentence p(x)t(x) and its associated
Message 7 of 8 , Jul 3, 2003
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<umistphd2003@y...> wrote:
>
> Your proof is a bit difficult to follow, so I can't comment much.
But it
> seems extremely suspicious that the Omega function should appear
like that
> in theorem 1.
>
> But surely someone's proved Perry's "conjecture" already...?
>
> Surely?!?
>
> Andy

I actually meant to print the following.

We are considering the sentence p(x)t(x) and its associated sum.
Define t'(x) as t'(x)={1 if t(x) = 0 and 0 otherwise. So now t'(x)
equals 1 iff (x,n)>1 and t(x) + t'(x) = 1 for all x in Sn.

Now p(x)t(x) is equivalent to p(x)(1-t'(x)) is equivalent to
p(x) - p(x)t'(x) for all x in Sn

If we sum over x then
Sum(p(x)) = Pi(n-3)-1 the odd primes p <= n-3

Sum(p(x)t'(x)) = Omega(n) - 1
because t'(x) equals 1 iff (n,p)>1 which implies that p|n but we
subtract 1 because since n is even 2|n and we only want the odd prime
divisors since p(x)=0 if x is equal to 2.

I believe that a simple formula for the Goldbachian partition
function will depend on Omega(n) but I am not ready to conjecture
such a result.

Thanks for looking over my post.
• So, what s next: Politically Correct Maths? Personally I *hate* PDF. Hmmm (muses to self: I wonder what Pytho and Archie used... PDF, HTML, UML, ....?) Fav
Message 8 of 8 , Jul 4, 2003
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So, what's next: Politically Correct Maths?

Personally I *hate* PDF. Hmmm (muses to self: I
wonder what Pytho and Archie used... PDF, HTML,
UML, ....?)

Fav counter example: consider a set X | X contains
phi and all sets which do not (by definition) contain
phi.

NOTES:

phi == {} == the empty set.

=========== (later froods, off to scarf down a
couple more hot dogs) btw, Barry White died :(

====================================== --30--
(the overt memo follows)

From: "Andy Swallow" <umistphd2003@...> |
Date: Thu, 3 Jul 2003 23:14:47 +0100
Subject: Re: [PrimeNumbers] Re: my elementary method
and Perry's /Goldbach's conjectures

Hey, its ok, I was only half serious. It's difficult
to set it out in plain
text anyway!

Andy

----- Original Message -----
From: "antonioveloz2" <antonioveloz2@...>
Sent: Thursday, July 03, 2003 10:52 PM
Subject: [PrimeNumbers] Re: my elementary method and
Perry's /Goldbach's
conjectures

> --- In primenumbers@yahoogroups.com, "Andy Swallow"
> <umistphd2003@y...> wrote:
> >
> > Could we pass a law?
> >
> > 'Anyone submitting proofs must typeset them
nicely'
> >
> > Please? It's a dream anyway...
> >
> > Andy
>
> I apologize. I will make a pdf file and a
mathematica program with
> the paper and post a link to them. I would also
like to state that I
--- Andy Swallow <umistphd2003@...> wrote:
>
> Hey, its ok, I was only half serious. It's difficult
> to set it out in plain
> text anyway!
>
> Andy
>
> ----- Original Message -----
> From: "antonioveloz2" <antonioveloz2@...>
> Sent: Thursday, July 03, 2003 10:52 PM
> Subject: [PrimeNumbers] Re: my elementary method and
> Perry's /Goldbach's
> conjectures
>
>
> > --- In primenumbers@yahoogroups.com, "Andy
> Swallow"
> > <umistphd2003@y...> wrote:
> > >
> > > Could we pass a law?
> > >
> > > 'Anyone submitting proofs must typeset them
> nicely'
> > >
> > > Please? It's a dream anyway...
> > >
> > > Andy
> >
> > I apologize. I will make a pdf file and a
> mathematica program with
> > the paper and post a link to them. I would also
> like to state that I
> > do not claim to prove Goldbach's conjecture.
> >
> >
> >
> > Unsubscribe by an email to:
> > The Prime Pages : http://www.primepages.org/
> >
> >
> >
> > Your use of Yahoo! Groups is subject to
> http://docs.yahoo.com/info/terms/
> >
>
>

=====
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