Examining a Prime Heavy Line

I wanted to look at a line that had a lot of primes on it to make sure there wasn’t some sort of pattern particularly with the ones that are not prime. The line that I used was the (p=73 x=4 c=8). I first calculated the base quadratic:

$4n^2+4n-7$

Again it has a c term which is no surprise. It hits zero not on a whole number and not on a 1/2 like mine did. It was some long decimal number.

I also looked at all the factors and other than it involves a lot of 7 factors I don’t see a pattern.  I think this is largely due to the fact they we are missing the 2 and  3 factors. In checking we also don’t have any 5 factors on this line. That also helps explain the high number of primes found here.

Below is a picture of the line. The purple squares are primes and the red squares have 7 as a factor.

I looked at the (p=421 x=10 c=8) line at it has no 7’s on it but still has a lot of primes. It does have a lot of 5’s and 11’s.