Wikipedia: Goldbach's Conjecture

Dass ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann

That every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.

Rationale:

We want two prime numbers that add to n. The only even prime number is 2. 2+p cannot add to n because n is even and p is odd. 2+2=4 is the only exception, so we'll only consider n>=6.

Start by finding all odd number pairs that add to n. If we find those then we can rule any that aren't prime.

	3	5	7	…	n-7	n-5	n-3
+	n-3	n-5	n-7	…	7	5	3
=	n	n	n	…	n	n	n

You can see that the right side of the … is just a flipped copy of the left side. 3 + (n-3) is the same thing as (n-3) + 3.

So we can just find the halfway point, and chop off the rest.

Let n = 34

3	5	7	9	11	13	15	17
n-3	n-5	n-7	n-9	n-11	n-13	n-15	n-17

Then we can color any primes green.

3	5	7	9	11	13	15	17
31	29	27	25	23	21	19	17

And if both values in a column are prime, turn them purple.

3	5	7	9	11	13	15	17
31	29	27	25	23	21	19	17

Observations

Looking at n, there are three possible ways (Q,R,S) to predict a goldbach pair in n+2 (the next even number)

Q: q+p = n+2, q and p are prime and less than n.

* note that the ? cells here could be prime or nonprime, either way the left pattern in n will create a prime pair from p and q in n+2

R: p = (n+2)/2, c is prime. iff n is divisible by 4.

if n is divisible by 4, the bottom right value will become the full right column.

S: n-1 is prime.

Even if Q and R aren't true, n+2 could still have a goldbach pair if the new bottom left value (NEW = (n+2)-3) happens to be prime.