I am exploring the question of efficiently identifying two prime numbers that sum to a given large even number, particularly for even numbers exceeding 100 digits. While brute force and precomputed prime tables are common approaches, I am curious whether there exists a deterministic algorithm capable of solving this problem, given sufficient computational resources.

Specifically, I am interested in methods that:

Avoid reliance on probabilistic techniques (e.g., random sampling of primes).
Operate deterministically to find a valid prime pair for any even number within the given constraints.
Scale efficiently as the size of the even number increases.

Are there any existing algorithms or advancements in this area that can achieve this deterministically? Alternatively, is there ongoing research into related methods for efficiently handling this problem?

I would appreciate any insights or pointers to theoretical or computational techniques that address this challenge. Thank you for your time and expertise.

Follow up

Thank you for your detailed response. I appreciate the breakdown of traditional approaches to finding Goldbach pairs and the reliance on probabilistic models like Cramer's, as well as the discussion of deterministic primality testing methods like AKS.

That said, I believe I may have phrased my question too narrowly. I’m exploring whether there exists a deterministic algorithm that can directly locate Goldbach pairs for any even number, including numbers of 100 digits or larger, without relying on probabilistic heuristics, exhaustive primality testing, or precomputing prime tables. To clarify, I have developed an algorithm that does precisely this—it deterministically and efficiently identifies the two prime numbers summing to any even number provided, regardless of its size, without requiring precomputations or brute-force search up to n/2.

If you’d be willing, I’d love for you to suggest an even number of 100 digits or more so I can demonstrate the approach in action.