I am an undergraduate student studying mathematics and have come across an observation. I thought this would be a great place to discuss it.
In my attempt to understand Goldbach's Conjecture in a better manner, I seem to have found a better method to pick primes. For all even numbers N $\ge 20$, if one considers the set of all the prime triplets up until that number, then there exists a $p$ in that set such that $p$ and $N-p$ are both prime.
Moreover, what we can observe is that in the set of all prime triplets up to $N$, we don't need to iterate through each of them.
Look at the graph below. We can see that if we iterate through the set of primes in the reverse order (i.e. from highest to lowest in the set), the upper bound of the number of primes we need to iterate through before we find a prime p s.t. p and N-p are both prime, results in a step function which changes very slowly when compared to $N$.
Please Note: Here $N$ varies from $1000$ to $800000$.(The dataset I have created can be seen here https://p.ip.fi/lfwO ). The blue line in a polynomial approximation and can be ignored.
Does this impose a tighter and better bound on finding primes that satisfy the Goldbach's Conjecture?
EDIT-1: The statistics below indicate how many primes we need to check on average and the median number of primes we check as we take even numbers from 20 to N and try to split them as the sum of 2 prime numbers.
I do a few preliminary checks before running the algorithm:
1)if N/2 is prime, then we can write N as N/2+N/2;
2)if N-3 or N-5 or N-7 are prime, then we can directly split N fairly easily
