Prime Number Patterns

Prime Number Patterns

Let's write the prime numbers in base 2 "binary" format. I like to use the convention of least significant digit to the left, so for example the number 11 in binary is 1101, and this simply means..

11=1*(1)+1*(2)+0*(4)+1*(8)

Here are the first 20 prime numbers in binary..

2 01
3 11
5 101
7 111
11 1101
13 1011
17 10001
19 11001
23 11101
29 10111
31 11111
37 101001
41 100101
43 110101
47 111101
53 101011
59 110111
61 101111
67 1100001
71 1110001

Let's try a pattern algorithm. We'll define a replace operator r(n) as "whenever the pattern 01 occurs in the binary representation of n replace it with 101". Here's an interesting thing..

r(2)=5
r(5)=11
r(11)=23
r(23)=47

But the pattern does not continue, because r(47) is not prime.

Of course, this is just a simple example of pattern recognition. If you have access to serious computing power there are many ideas you can test!

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Content written and posted by Ken Abbott abbottsystems@gmail.com