Elimination Patterns

I have thought about primes off and on for a long time. One of the ways I looked at primes was by looking at which numbers eliminate what. It never really amounted to anything but when I saw the Ulam Spiral I wondered what would happen if I plotted the eliminations. So basically plotting what a sieve would do. The first was all those eliminated by 2. No surprise there, it came out as a checker board. It does force line to be on the diagonals as well as provide white space on either side of a line.

Ulam Sprial 2 Factors

The next set were the numbers eliminated by three.

Ulam Sprial 3 Factors

I wasn’t really surprised to see a pattern. The interesting bit are there are lines that thread their way through without hitting a divisible by three number. Also there are lines that hit alot of numbers that are divisible by three. Two examples of these lines are (p=21 x=2 c=8) and (p=73 x=4 c=8). The first hits lots that are divisible by 3 and the second hits none.

Ulam Sprial 3 Factors Line Comparison

Needless to say this is going to have a big effect on where we see the strong lines.This is also where I started noticing the exclusion lines. The exclusion lines are discussed on another website by H. Rudd. As he says on one of his pages I am glad that I didn’t find his site until later in my thought process as I would have gone in a different direction. (Note his spirals go in a different direction than mine) The (p=21 x=2 c=8) line has no primes on it and neither does (p=273 x=7 c=8). I had to write a program to investigate those but that will be another page.

The numbers with 5’s as factors also had a pattern and again some lines managed to miss them all.

Ulam Sprial 5 Factors

The numbers with 7 as factors also form a pattern and again there are lines that go through without hitting a number that is divisible by 7. Note however that the line I picked in the image below does hits quite a few that are divisible by 5.

Ulam Spiral 7 Factors

I also did an image up for numbers with 11 as a factor. It looks regular but I can’t see a pattern. Maybe it’s to diffuse and I need a bigger plot. Either way I decided to stop here.

Ulam Spiral 11 Factors

This is also where I got tired of hand drawing and cobbled up the images you are seeing. Basically I hand drew it out on a spreadsheet and took a screen shot. Then I overlaid the grid lines on a new layer. Each set of factors (2,3,5,7,11) are on an individual layer and were hand filled.  I also have a layer that are the primes.

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