I was arranging numbers putting them in different orders when I happened to build a pyramid and noticed that two columns of the pyramid seemed to contain all of the prime numbers. Not just some of them, but all of the first hundred primes at least.
Considering how rare I thought that might be, I'm bringing it here for consideration.
I began with the numbers 5, 6, 7 at top. Then going diagonal to the left for 5, I did 52, then 53, etc as the left side of the triangle. On the right I did 71, then 72, etc going down the right side of the triangle.
This means the top row is 5, 6, 7. The next row is 10, 11, 12, 13, 14. The next is 15, 16, 17, 18, 19, 20, 21... and so on.
If you look at the columns under 5 and 7 they contain ALL of the prime numbers.
Is there a way to create a proof that all prime numbers do NOT exist in these two columns?
The column under $5$ consists of all numbers of the form $6n-1$, and the column under $7$ consists of all numbers of the form $6n+1$. All primes $> 3$ are of one of these forms.