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_
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