Chalkboard #3 Prime Classes

This chalkboard discusses an observed relationship between prime numbers and finite sequence lengths of the Perrin sequence. It is central to my future discussions of the length of periodic points and the number of orbits.
1. Type 2 Primes
Consider an algebraic equation modulo 23 in an unknown integer x.
X2 = S mod(23)
Let x be a an integer greater than 0. The first few numbers x are 1, 2,3,4,5,6,7,8 and the corresponding squares are 4,9,16,25,36,49,64…
It is easy to see that 12 = 1 mod(23) 22 = 4 mod(23) and 32 = 9 mod(23)
Continuing; 42 = 16 mod(23), 52 = 2 mod(23), 62 = 13 mod(23), 72 = 3 mod(23)..
It can be shown that for x between 0 and 23, the numbers S are in the set of integers
S ={0,1,2,3,4,6,8,9,12,13,16,18} only. Given any prime e.g 29 we see that
29 = 6 mod (23) = 112 and 6 is a member of the set S. Large primes eg p= 587 when converted to
mod (23) may be in the set S : 587 = 12 mod(23) = 92 mod(23)
The sequence of primes that are squares mod(23) are given in OEIS A191021.
2,3,13,29,31,41,47,59,…..
Chalkboard3_Prime_Classes