Appendix to Chapter 21

The divisibility of polynomials  to G(x) = x^3+Bx^2+Cx-D was derived in Chapter 21.  In this appendix the divisibility is extended to conjugates of G(x) and other higher degree polynomials.   A general conjecture is made regarding all cubic irreducible polynomials of negative discriminant.

Appendix to Deducing Polynomial Division Algorithms_

Chapter 21 Deducing Polynomial Division Algorithms Using a Groebner Basis

A computerized algebra algorithm is used to find polynomials and the integer coefficients a, b and n for which the polynomial F(x) = a x2n + bxn + Dn – (Bx2+Cx+D)n is divisible by G(x) = x3 +Bx2+Cx+D.  Consequently, x is any of the real or complex roots of G such that G(x) = F(x) = 0.  Also, the coefficients a, b are integers derived from the integer sequence associated with the cubic polynomial G. Higher order polynomials divisible by G(x) are also derived using a Groebner basis. It is shown that new recurrence relationships can be generated from the Groebner basis.

Chapter 21— Deducing Polynomial Division Algorithms

Chapter 20_Integer Sequences, Discriminants and the Dedekind Eta Function

In this Chapter I will discuss the complex eta function and how the eta quotient can be used to find the real solution of several irreducible cubic polynomials.  For some particular prime and negative binary quadratic discriminant, the eta quotient can be used to find primes which split the irreducible polynomial mod p. Once these primes are found all irreducible polynomials of degree 3 can be converted to integer sequences.  The magnitude of the period of these sequences is further discussed.

Integer Sequences, Discriminants and the Dedekind Eta Function_

Perrin Sequence Lengths – Originally generated by Christian Holzbaur

The period of Perrin (0,2,3,2,5,5,…, A001608) sequence mod n. A Mathematica program is found for Perrin Periods in OEIS A104217

Board20a_Perrin Sequence Lengths

Chapter 19 Geometry of the Perrin Sequence

The plastic number is a mathematical constant which is the real irrational solution of a monic cubic equation.  It is also the limiting ratio  Pn/Pn-1 of the Perrin and Padovan sequence of numbers.  Equations of degree 3 are not solvable by plane geometric projections using a compass and ruler.  A 3D solution to the problem is possible using paper folding or Origami techniques.  This Chalkboard demonstrates a construction of the plastic number and also its positive and negative powers.  The angle of trisection also contains information on powers of this irrational number.  It is also shown that the Perrin sequence and Padovan sequence are generated from the unit measure and powers of the plastic number.

Geometry of the Perrin_ and Padovan_ Sequences





Chapter 18 -A General Algorithm for Restricted Partitions

A general algorithm is discussed based on the total number of bonding and non-bonding partitions into parts of arbitrary integer modulus. The  bonding partitions introduces the Padovan sequence from the Perrin sequence of numbers. The Padovan sequence is shown as a representation of the partition of integers into odd parts. A method for calculating the Padovan number expressing each  basis representations as bonding and non bonding integers mod 5 is developed.

Chapter 18 _A General Algorithm and Perrin and Padovan Sequences






Appendix 17_Observations of Restricted Partitions

In this Appendix to Chapter 17 I will discuss various observations on restricted partitions of the Rogers- Ramanujan Identities.  An integer N can be partitioned into parts 2 mod 5 and 3 mod 5 according to the second identity.   The number of parts of length or depth k is found to depend on both the modulus of k and the modulus of N. Some congruences are also given. Generating functions for each basis representation  can be derived based on simple rules.

Appendix to Chapter 17_Some New Observations for the Restricted Rogers-Ramanujan Identities




Chapter 17 Perrin’s First Sequence and Other Isomorphic Recurrence Sequences

Perrin’s short Query in 1899 mentions two sequences; the first a Fibonacci sequence and the second the famous Perrin sequence.  This Chapter looks at their origin and discusses the relationship between various Fibonacci and Perrin sequences.  The objective is to begin a discussion in number theory and combinatorics on arranging numbers into various integer partitions. The size and number of these partitions are found and can be reduced to sequences of integers.

Perrin’s First Sequence and Other Isomorphic Recurrence Sequences_


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_