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

Chalkboard #2 Finite Sequences

OEIS A104217
Finite sequences are periodic sequences. Rules are used to define these finite sequences.
A musical example is given as follows:
A song is being composed using only 2 notes (C and D). After 2 notes are played the 4th note is played based on the following rules:
If CC played then C, If DD played then C , if DC or CD played then D.
Starting with the 3 notes DCC the song will appear as the following sequence of notes:
DCCDCDDDCCDCDDDC
Notice that after 7 notes the sequence repeats ad infinitum.
Note: If the second note was E instead of D in the example above the sequence above would end as Beethoven’s Fifth Symphony begins!
Given the infinite Perrin Sequence 3,0,2,3,2,5,5,7,10,12,17,22,29,… convert each number to modulo 2 which will give a series of 0’s and 1’s:
1001011100101….
Chalkboard2_Finite_Sequences

Perrin Chalkboard

 

Welcome to the Perrin Chalkboard!  I will be presenting in this blog a series of chalkboards which discuss interesting properties of the Perrin sequence and related integer sequences. This blog starts as a simple discussion of the Perrin sequence (the original mention by Lucas in 1876 and Perrin in 1899). It is found that an immense amount of research on the associated elliptic curves has occurred over the last 115 years. The Perrin sequence ties together much of the mathematics discussed today as algebraic number theory and modular functions. It is also integral to the discussion of Fermat’s Last Theorem conjectured in 1637 but proved by Wiles in 1994.

Theorems will be presented without proofs. I think the subject matter will appeal to those interested in the properties of integer sequences, elliptic equations, and graph theory. Many sequences from OEIS (On-line Encyclopedia of Integer Sequences) will be discussed, uncovering hidden or less obvious properties.

The primary subject matter in this blog covers the properties of integer sequences. However, it is not until Chapter 17 that I cover the subject matter of the short paper published by Perrin in 1899.  This chapter then introduces the subject of integer partitions, followed by some geometric applications of the Perrin sequence and then turns to division algorithms derived from general properties of cubic equations and associated integer sequences.

I encourage any comments or suggestions to the chalkboard subjects.

Although the pdf files are freely provided, if you are interested or have questions regarding any chapter please feel free to contact me.

Richard Turk
March 2015

Updated July 2017

Click on the pdf to open the first chapter.. or  continue scrolling to find the latest Chapter and pdf!

Board1_generation