- 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
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) ]
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.