Sharpening the Goldbach conjecture

I like to think on famous mathematical conjectures, because these are real problems and their solution may be rewarded by fame. Even if I do not come to any solutions, I get some experience that is probably more valuable than the solution of made-up exercises. For example, I have already spent time trying to prove the Hadwiger conjecture in graph theory (including the Four-color conjecture, which is called theorem since), the Collatz conjecture, the Twin Prime conjecture, Goldbach's conjecture, Beal's conjecture, the Riemann hypothesis and the P versus NP problem, and as well as some own problems I came across during recreation.

From all of these conjectures, the Goldbach conjecture was the most interesting to me so far. This says that for every even number greater than 2, there exist two prime numbers such that the sum of these prime numbers are equal to the even number. You can probably see a picture about examples here for even numbers 4 to 500:

If we choose the prime numbers like I did on this picture, so that the prime numbers should have the least difference, we can notice that these prime numbers are all between the 1/4 and 3/4 of the original even number, and as the even numbers grow, the smaller prime numbers are not needed any more. This is actually my new conjecture that I called the Sharp Goldbach Conjecture, and I have checked it by computer for "small" even numbers (as far as I remember, some millions). By the way, this name for the new conjecture is probably not good, as I guess mathematicans call something sharp if it cannot be sharpened further. I cannot call it Strong Goldbach Conjecture either, because the Strong Goldbach Conjecture is the Goldbach conjecture itself for historical reasons, and Goldbach's Weak Conjecture is about any odd number greater than 5 being the sum of three prime numbers. I think I will call it my Sharpened Goldbach Conjecture.

Otherwise, as far as I remember, I showed to myself that the Goldbach conjecture itself is also equivalent to a conjecture that deals with only prime numbers greater than an arbitrary number, provided that the sum is also greater than a specific number (dependent on the arbitrary number). As far as I remember, I transformed the statements telescopically, and there was a square of a prime, but I do not remember exactly how. I tried to reproduce the proof here, while writing this blog post, but it turned out that I could not. So... this is your homework! (Just kidding.)

After these insights, I tried to prove the conjecture by using numbers NOT divisible by little numbers / primes instead of using prime numbers. This attempt did not succeed, but I realized that this is looking like Sieve Theory, by which other mathematicans may try to prove the Goldbach conjecture or the Twin Prime conjecture. So I think there might have been some chance for the proof, but I had not only mathematical problems to think on, but also real-life problems in the world.

Summary: My Sharpened Goldbach Conjecture says that every even integer greater than 3 can be written as the sum of two prime numbers between the 1/4 and 3/4 of the even integer. Sharpening conjectures might help in knowing their nature.