A simple estimate of the number of Goldbach pairs for N

Abstract

Using the fact that the number of combinations $p_{1}$, $p_{2}$, where $p_{1}$ and $p_{2}$ are odd primes, with $p_{1} \leq p_{2}$ and $p_{1} + p_{2} \leq 2N$ is equal to the total number of Goldbach pairs for all the even integers from 6 to $2N$, we derive in a simple way an accurate estimate of the number of Goldbach pairs for $N$