Let's make a 2 by 2 crosstabs for the number of odd pairs(P,Q) which add to

2 * N.

And assume that primeness is independent of the adding to 2*N. (Which is a

separate conjecture.)

Q odd composite Q odd

prime Q odd total

P odd composite ( N - N/log(N))^2 [

N/log(N)] * [N - N/log(N) ] N * [N - N/log(n) ]

P odd prime [ N/log(N)] * [N - N/log(N) ] ( [

N/log(N)] * [N - N/log(N) ] )^2 N/log(N)

P odd total N * [N - N/log(n) ]

N/log(N) N^2

This suggests that the number of ways that primes add to 2N increases

as N increases.

Thus I propose two companion conjectures to the Goldbach conjecture.

(1) The number of ways in which an even number M is the sum of two primes

increases as M increases.

(2) The number of prime pairs that add to M is approximately what would be

expected if adding to the even number

M is independent of primeness.