On Number Theory and Goldbach's Conjecture

Question

Good day. My question is: Does every even number have the form $n=p+ 2+q$, or $n=p- 2+q$ with $p, q $ prime numbers, such that $p\pm2$ is prime number? Up to $n = 1000$ I know that it is true, and it isn't equivalent to goldbach's conjecture.

Answer 1

(1) With the new hypothesis that you added that $p \pm 2$ must be a prime, your conjecture reduces to Goldbach's Conjecture. Do you see why? (and ignoring the fact that it presupposes an infinitude of twin primes.)

(2) If you remove this new hypothesis and leave the problem as you originally stated it, then you would also have the following theorem: "My conjecture is true if and only if Goldbach's Conjecture is true.'' And the reason for this is as follows: Since Goldbach's Conjecture is true, as a commentator pointed out earlier, for every even number up to $4 x 10^{18}$, then choose an even number $n$; then, say, subtract 2 from it. It will be a number which satisfies Goldbach's Conjecture. Thus, take a pair of primes $p, q$ satisfying Goldbach's Conjecture (there may be more than one such pair) whose sum is $n - 2$, and then add 2. This gets you back to your $n$. A similar argument follows if you choose an even number $n$ and then add 2 it.

So, I guess it could be said that your original conjecture is true for all even numbers up to $4 x 10^{18} - 2.$