This appendix updates the theory of binary sequences from the results discussed in Chapter 13. In that chapter the Perrin sequence was found to produce a period 14 binary pattern from the Sigma orbit defined in OEIS A127687. An enhanced formula for the sigma orbit is developed for use with sequences from general cubic polynomials. The results show 6 classes of binary sequences are obtained from the the ring of polynomials of degree 3. The rules for class membership are defined.
For Perrin pseudo-primes the period 14 binary sequence predicts that pseudo-primes can occur at 2, 4 and 8 mod 14. To date for numbers <10e14 only PPP(3)= 2 mod 14 and PPP(5) = 4 mod 14 have been confirmed. Can it be determined if PPP(3)*PPP(5) = 453371887665796 = 8 mod 14 is a Perrin pseudo-prime?
Appendix 13d to Perrin Pseudoprimes
The Perrin Conjugate and the Laguerre Orthogonal Polynomial
The exponential expansion of the Perrin conjugate leads to a series like the exponential generating function for the Laguerre polynomial. This orthogonal polynomial can be used to expand any polynomial in a series of Laguerre polynomials. A summation series has been developed for the classic orthogonal polynomials. Integral representations are derived using the orthogonality of the Laguerre polynomial to find monomial terms of Legendre, Hermite and Chebyshev polynomials in terms of the Gamma function. Expansions can also be easily derived for these classical polynomials using the confluent hypergeometric function. The connection of these polynomials to symmetric functions is also demonstrated.
The Perrin Conjugate and the Laguerre Orthogonal Polynomial (2)
In the previous chapter P2(x,n) was found to be a polynomial of degree 3n and divisible by a cubic polynomial G(x). In this chapter the division is defined in the finite field of the discriminant of G(x). Limits are placed on the degree 3n when the polynomial is to be completely factored in the field.
The decomposition of an N dimensional space into invariant sub-spaces is demonstrated using Groebner basis. Similar matrices are derived from the characteristic polynomials P2(x,n) and represented by symmetric geometric shapes.
Factoring P2(x, n) over a Finite Field
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_
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
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
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
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
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
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_