At this point I wrote a simple program that took the p,x,and c of a line and calculated it out as far as I specified. As it went it also calculated the factors of each number. The first line I ran was the (p=21 x=2 c=8) line. The program took it out to 31621 iterations and a final number of 4,000,183,005. There were no primes found. This wouldn’t prove that there never would be. I wanted to know why. The first thing that I noticed was that it ran parallel to the squared odds line and was 4 away from that line.
I then noticed that some of the factors at the far end of the scale were a pair of large numbers that were exactly 4 apart. So I looked back at the beginning and saw that the factors of 21 are 7 and 3 which are four apart. Looking at the next number 45 its factors are (3,3,5) which when combined right make 5 and 9. Again 4 apart and each is 2 more than the 3 and 7 from the previous number. Below is a spreadsheet. Iteration is just a count along the line. Number is the current calculated number along that line. Factors are the computer generated factors of the number. Working is were I was combining the factors trying to make the pattern. Calculated is using the fact that each number in the working columns is 2 more that the line above it. So the formula for the first column under calculated is 3+2*iteration and the second column is 7+2*iteration. I wanted to see if this was working so I extended it out and checked to see if it matched. As you can see it does.
I wanted to make sure that it continued so I did the calculations as above with the last numbers from the output of the program. And again it did.
Next up is does exclusion line (p=273 x=8 c=8) follow the pattern.