Chapter 15 Calculating the nth Term in the Perrin Sequence when n is Prime

Based on the results of Chapters 13 and 14 an equation is developed to calculate directly the value of Perrin(N) when N is a prime number and where Perrin(n) is given from the sequence P(n) = P(n-2) + P(n-3) with P(1) = 0, P(2) = 2, P(3) = 3.
A modified incomplete Beta function is derived to calculate each term of the Sigma1 orbit but these terms can be further simplified to a short sequence in (A,B) where A and B have been previously defined. The Perrin sequence can also be expressed as a hypergeometric function.

Calculation of the Perrin Sequence with a Modified Beta Function_

Chapter 14. Perrin Prime Distribution Theorem

Perrin primes and composite numbers of these primes are shown to distribute uniformly in a 14 column by 14X row grid. Dirichlet’s theorem on arithmetic progressions states that there are infinitely many primes of the form a+nd (a congruent modulo d). This Chapter discusses the case when d=14. There are 6 progressions of primes modulo 14 as predicted by Euler’s totient function. A prime divisor function f(x,y,N)=0 is derived to factor vertical entries containing integer N. An algorithm can be programmed to solve for the zeros of the prime divisor function.


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



Chalkboard #12 Elliptic Curves over the Rational Number Field

This chalkboard should have been an appendix to Chalkboard #3 on Classification of Prime Numbers. I previously defined three types of primes which were used to classify the period length of the Perrin sequence modulo (p).

If we start to look for rational solutions mod (p) to the elliptic curve defining the Perrin sequence the reason for this classification becomes apparent.

The ideas for this chalkboard came after reading “Fearless Symmetry” by Avner Ash and Robert Gross (Princeton University Press) 2006.
Board12_Elliptic Curves Over the Rational Number Field

Chalkboard #11 Maximal Independent Sets, Cycles and Spanning Trees

In graph theory an independent set is a set of vertices in a graph in which no two vertices are adjacent so no edge connects these two vertices. The maximal independent set (MIS) is the largest set of independent sets for a graph. An MIS is not a subset of any independent set.

The number of maximal independent sets for an n-cycle or an n-gon graph can be found from the Perrin sequence of numbers.

Reference: R. Bisdorff and JL. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, J. of Integer Sequences, Vol. 11 (2008).