Pattern to the Exclusion Line Locations

So I needed to know where the exclusion lines were.  I wrote a program that went down a line and looked at each line going off it in a particular direction.  The first line that I used was the (p=7 x=1 c=6) line. I checked each line going off in c=8 direction. To speed things up since I was looking for line without any primes the program moved onto the next line as soon as it found a prime.

Ulam Spiral Exclusion line Locations

Below is the spreadsheet that I generated from the output of that program. As you can see it found the lines that we have already talked about. Now due to a happy accident it was reporting the x value as one less than it actually was. The first thing I spotted was all the prime numbers in the x location along the first line.

Exclusion Line Location

Now that almost caused me to miss something but instead it led to more questions. Using the correct values for x there is a clear pattern.  The locations are growing in a regular pattern. So count is just used in the calculation column. X is the location of the exclusion line on the first line. The next column is the current X minus the previous X. Difference is the same idea calculated on the previous column. Calculated uses those number to calculate the  next X based on the previous X. The formula is

I probably should have moved the count down by one for clarity but that was not how I was thinking.

Exclusion Line Location_

So at this point I went and read H.Rudd’s website thoroughly and while he came at this very differently there are a lot of parallels to what I found. He discusses describing these lines as parabola with quadratic equations. He was working with different lines but when I tried it with these lines it also worked.  The (p=273 x=8 c=8) maps to the parabola described by the quadratic equation:

The (p=1333 x=18 c=8) line maps to the parabola.


Why do are these exclusion line locations frequently one greater than a prime number?

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