- Hello all,

I've been working on the twin prime conjecture for about five years

and would like to share some of my ideas (no proofs as yet though,

sorry!)

The moebius function: M(n)

M(1)=1

M(n)=0 if n is divisible by m^2 for some m>1

M(n)=(-1)r if n is the product of r distinct primes,

The Mangoldt function: A(n)

A(n)= log p if n=p^k, k>0

0 otherwise

a function that defines the primes: F(n)

F(n)= M(n)*A(n) gives 0 if n is square-divisible or divides a product

of primes, or log(1/n)=-log n if and only if n is prime

therefore, F(n)*F(n+2) is equal to log n * log (n+2) if and only if n

is the first prime in a twin pair. (The second prime n+2 would not

produce a nonzero result since n+4 is not prime, and n>3) So then,

sum n<=x(F(n) * F(n+2))/(n * log n * log (n+2))

would sum the reciprocals of the first prime in a prime pair (by

using the error term (2*c)/log n (c=.6601618158 the twin prime

constant) I've supposed this limits to about 1.05893... (*)

I'm using (2*c)/log n instead of Prof Brent's term (4*c)/ log n

because I've summed this using one term in the pair instead of the

usual two.

The constant I marked (*) appears to be constructed out of a ratio of

prime power products:

product n=1, infinity, 1-M(n)*n^-2 divided by

product odd primes 1-(p-1)^-2

(The denominator being the constant c=.66016...) This appears to be

the same as the sum of reciprocals of the twins defined above, in

terms of a limit (I dont think this assumes infinitely many twins

since Brun's constant is the limit of 1/p+1/(p+2) regardless of the

number of terms in the sum). Any one like to prove this? Or, if I am

wrong please show me where. (you might complain about using the

moebius function in a product...!) If this is proven, what will it

say about the total number of twin pairs?

Thanks,

from Guy H Bearman > I've got a bit of a fascination at the moment for composites (c) of the form

Not particularly. The first two examples are far too small to draw any

>

> c = p^2

>

> One of the things I seem to be observing is that there seems to be twin

> primes in close

> proximity to composites of the above form.

>

> eg.

>

> 5^2 = 25 17,19 and 29,31

>

> 7^2 = 49 41,43 and 59,61

>

> 317^2 = 100489 100391,100393 and 100517,100519

>

>

> Is this significant?

reasonable conclusions from. It seems strange that you suddenly have to

jump from 7 to 317, but maybe there are other examples in between, I

can't be bothered to check.

According to the prime number theorem, the average distance from a prime

p to the next prime is about log p. Suppose now that p is part of a twin

prime pair, and that q is the first of the next twin prime pair. Then

the expected distance between p and q is roughly log^2 p instead. So the

twin primes are expected to be fairly common, and certainly a lot more

common than numbers of the form p^2. So it's not surprising that you may

well find twin primes "fairly close" to some prime squares.

Andy