## Re: A goldbach conjecture for twin primes

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• ... also ... sort of ... There are ... is not ... I think primes satisfy GC not because of their primality but because of their random enough
Message 1 of 8 , Dec 2, 2001
--- In primenumbers@y..., "Harvey Dubner" <hdubner1@c...> wrote:
> There has been so much discussion about the Goldbach conjecture and
also
> about twin primes that I can't resist adding a conjecture which
sort of
> combines them.
>
> Define a t-prime to be a prime which has a twin.
>
> Conjecture: Every sufficiently large even number is the sum of two
> t-primes.
>
> More exact: This is true for all even numbers greater than 4208.
There are
> only a few even numbers less than or equal to 4208 for which this
is not
> true.
>
> Harvey Dubner

I think "primes" satisfy GC not because of their "primality" but
because of their "random enough" distribution. In fact, with ever
larger even numbers, ever less percent of primes is needed to satisfy
GC. With my computer power, less than 1% of the primes would suffice.

Defining g(i) as the product of (1-2/p) for all prime p<=sqrt(n), the
number of Golbach pairs for the even n is almost (n/4)*g(i).
Similarly, the number of t-primes between p^2 and (p+2)^2 is almost
2p*g(i), which means t-prime distribution is far denser than what is
needed to satisfy GC. In fact, every set of random odd numbers with a
spacing of the order n^1/3 is enough to make all evens with their
paired sums, while t-prime spacing is of the order log(n)^2.

For the same reason, the GC-like conjecture for the triplet-primes is
also eventually true.

Kaveh
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