- Jul 3, 2003The 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.

Please notify me with any questions or comments. - Next post in topic >>