N >= 6 would have to be the sum of two odd primes. An odd number x is prime with probability about 2/ln x. X and n - x are two primes with probability about 1 / ln^2 n, so you need to examine about ln^2 n pairs of numbers to find a pair of primes.

So that shows that creating a sieve of size n is very inefficient. It takes n steps when you usually need just ln^2 n. So create one sieve for the numbers c x ln^2 n and one for the numbers n - c x ln^2 n to n. Experiment to find a “c” that usually finds a solution, and if it doesn’t work then repeat with 2c instead of c.

Now if you wanted to check all even 6 <= N <= 10^9 then you would create one large sieve.

N ≥ 6 would have to be the sum of two odd primes. An odd number x is prime with probability about 2/ln x. X and n - x are two primes with probability about 1 / ln² n, so you need to examine about ln² n pairs of numbers to find a pair of primes.

So that shows that creating a sieve of size n is very inefficient. It takes n steps when you usually need just ln² n. So create one sieve for the numbers cxln² n and one for the numbers n - cx ln² n to n. Experiment to find a “c” that usually finds a solution, and if it doesn’t work then repeat with 2c instead of c.

Now if you wanted to check all even 6 ≤ N ≤ 10⁹ then you would create one large sieve.

N >= 6 would have to be the sum of two odd primes. An odd number x is prime with probability about 2/ln x. X and n - x are two primes with probability about 1 / ln^2 n, so you need to examine about ln^2 n pairs of numbers to find a pair of primes.

So that shows that creating a sieve of size n is very inefficient. It takes n steps when you usually need just ln^2 n. So create one sieve for the numbers c x ln^2 n and one for the numbers n - c x ln^2 n to n. Experiment to find a “c” that usually finds a solution, and if it doesn’t work then repeat with 2c instead of c.

N >= 6 would have to be the sum of two odd primes. An odd number x is prime with probability about 2/ln x. X and n - x are two primes with probability about 1 / ln^2 n, so you need to examine about ln^2 n pairs of numbers to find a pair of primes.

So that shows that creating a sieve of size n is very inefficient. It takes n steps when you usually need just ln^2 n. So create one sieve for the numbers c x ln^2 n and one for the numbers n - c x ln^2 n to n. Experiment to find a “c” that usually finds a solution, and if it doesn’t work then repeat with 2c instead of c.

Now if you wanted to check all even 6 <= N <= 10^9 then you would create one large sieve.