Additional Thoughts on the Goldbach Conjecture

The Goldbach Conjecture states that all even numbers greater than 2 can be expressed as the sum of two primes. With modern computers, no counterexamples have been found for numbers as high as 4x10¹⁸ (Goldbach's conjecture - Wikipedia). The lack of finding a counterexample is not a proof; however, it is an indication the conjecture is true.

The table shown below gives the sums of numbers 1 through 40. Highlighted rows in blue and columns in green are the prime numbers. Wherever these colored rows and columns intersect would give a number that is the sum of two primes. 

Casually walk through this table from the upper left corner down toward the lower right corner and look for even numbers greater than 2 that occur at the intersections of the green columns and blue rows (primes). No counterexamples can be found in this small sample. Another way to review is to observe that each even number can be found along multiple cells lying along diagonals (two such diagonals are highlighted in yellow for 34 and 58). To find a counterexample, one would have to find a diagonal that traverses through the table without landing on an intersection of a green column and blue row. In the crowded table above, it seems this would be a difficult task.

Now, we know the density of primes drop as numbers get higher. Look at a continuation of this table - just showing the portion 1000 - 1020 - where the density of primes is much lower.

Two diagonals are highlighted for the even numbers 2022 and 2030. Within the range of the table, 2022 crosses through some intersections meaning it is a sum of two primes. The diagonal for 2030 doesn't cross through any such intersections. This limited snapshot is not actually finding a counterexample because the rows 1-999 and columns 1-999 are excluded and it is within these excluded zones where the diagonal for 2030 crosses through intersections of prime numbers. My purpose in showing this portion of the table at higher numbers is to show that these diagonals can pass through portions of the table without coming across intersections of primes. A proof of the Goldbach conjecture would arise if one could prove that the diagonals of even numbers must cross the intersections of primes.

See here for an interactive version of the graphics above in The Wolfram Demonstration Project.

Link to prior post on Goldbach Conjecture.