Preface to Perrin Sequences My Chalkboard


My interest in Perrin sequences began after reading a back issue of Scientific American. In the June 1996 issue, Ian Stewart discusses a divisibility conjecture for primes. After examining periodic number patterns produced by the Perrin sequence modulo m (m is an integer > 1), I came across an interesting pattern: the sequence length P(m) depends on the type of prime, type 1 when P(m) is divisible by m2-1 and type 2 when P(m) is divisible by m2+m+1.

At the time I could not find any obvious reference to this in the literature so I began a long exploration into the properties of integer sequences. The chalkboard chapters are a systematic discovery of some interesting mathematics of integer sequences.

Chalkboard #1 discusses the origin of the infinite sequence from the roots of an associated elliptic equation.

Chalkboard #2 then discusses how finite sequences are generated and the classification of prime numbers to describe the properties of three types of sequences modulo m.

Matrix representation of the sequence and properties of the eigenvalue matrix to find solutions to modular equations are discussed in Chalkboard #3.

Perrin sequences modulo m are classified into equivalence classes based on sequence length, modulus and an initial generating integer 3-vector. (Chalkboards #4 and #5).

The growth of periodic orbits of various equivalence classes is shown in Chalkboard #6. An attempt is made to find all equivalence classes for a given modulo m.

It was realized that since 4th order polynomial equations can be reduced to 3rd order equations by change of variable, higher order equations can also generate Perrin-like sequences. An interesting formula is derived to find period length and orbit count of 4th and 5th order sequences.

In Chalkboard #8 the Weierstrauss form of a 4th order polynomial is used to find sequences from other elliptic equations. These sequences are a series of rational fractions instead of integers. A finite Fourier transform is used to find the period of these sequences.

In Chalkboard #9 the generating function to express the Perrin sequence as a power series is presented.
This leads to a discussion of the Zeta function which relates the period and orbit count through a power series function. The generating and zeta functions are related by an integral transform.

Chalkboard #11 is a detour in maximal independent sets (MIS) showing an application of the Perrin sequence to graph theory. Zeta functions of the MIS are presented.

Chalkboard #12 culminates in seeking rational solutions mod p to the elliptic curve. The Birch Swinnerton Dyer Conjecture appears to link previous concepts of prime classification, rational solutions and irreducible characteristic polynomials.
Some observations to isomorphic modular forms are also discussed.
Irreducible representations of elliptic curves and associated modular forms are important concepts required in proving Fermat’s Last Theorem.

Perrin Pseudoprimes are discussed in Chalkboard #13. The sigma orbit equation introduced in Chalkboard 11 is used to prove various Perrin pseudoprimes.

A prime distribution theorem and calculating prime terms in the Perrin sequence are discussed in boards 14 and 15. Several addendum a have also been included.

Chapter 17 and 18 discuss other related sequences such as the Rogers-Ramanujan identities and some applications to generate the Padovan Sequence.

I added Chapter 19 to explore a geometry of the Perrin sequence and how paper folding techniques can be applied to this sequence.

In future Chapters, I plan to discuss modular functions and introduce class field theory.

Richard Turk
Plymouth MA
May 2015. Updated September 2016




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!