Finally, I want to take a look at what we can know about the distribution of semi-primes on this grid. Here it is again as a reminder, with the primes coloured pink and the semi-primes yellow (other colours indicate numbers with more than 2 prime factors).
So what do we know about the primes and semi-primes on this grid.
1. There is an infinite number of primes in the central axes. I think Euclid's proof of the infinitude of primes might also suggest that there is an infinite number in each of these axes but I'm not 100% sure of that - I'll assume it is true for now and try running through the logic in another post.
2. Either way there is an infinite number of semi-primes on the grid. (This is obvious anyway, since each prime number will generate an infinite chain of semi-primes when multiplied by each of the infinite chain of primes.
3. Each prime on the central axes will generate an infinite, intermittent horizontal or vertical line of semi-primes stretching across the grid. This will hit an infinite number of diagonals.
What we don't know:
1. Is there an infinite number of semi-primes on the twin prime diagonal? If so the twin prime conjecture is true. The observations above suggest why this might be true, but they aren't proof of it.
2. Is there an infinite number of semi-primes on every infinite diagonal on the grid? If so, Polignac's conjecture is true (this refers to the larger grid as seen in the second and third image on this page).
3. Is it impossible to take a diagonal from one axis to the other (or from an axis to the diagonal NW-SE boundary - see previous post for a clearer explanation) without hitting a semi-prime. If so the Goldbach conjecture is true.