# Connecting the Angled Exclusion Line to the Square Odd Line

I found the connection between the square odd line and the angled exclusion lines that we have been looking at. A while back I was looking at prime numbers and noticed that it seem to be that I found primes close to the square of prime numbers. In testing that I found that

$p^2+2$

was never prime. I shared that and other info with a friend that is a Math Professor and he told me that that was well known and showed me a proof of it.

To prove it, note that any prime p other than 3 cannot be divisible by 3, so it is either 1 more or 1 less than a multiple of 3 (i.e. its remainder after dividing by 3 is either 1 or 2). Representing p in each case as either p=3m+1 or p=3m-1, respectively, and then substituting that expression into p^2+2 and expanding makes it clear that this must be a multiple of 3.

I sent him an email last night telling him of this website and remembered that fact and wondered what would happen if I used odd numbers. So we define odd as

$2x+1$

Then looking at the (p=21 x=2 c=8) line which is 4 off of the square odd line so it would be

$p^2-4$

We then substitute the 2x+1 in and get

$4x^2+4x-3$

After spending way too much time remembering how to factor I came up with

$(2x-1)(2x+3)$

Thus this can be generically factored and thus always be not prime. It also nicely shows the difference of 4 between the factors we saw earlier.

I also looked at the (p=273 x=8 c=8) which is -8 from the square odd line. It also work with the quadratic being

$4x^2+4x-15$

That factors to

$(2x-3)(2x+5)$