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.
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.
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.
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
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.
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.
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.
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.
This chapter continues the discussion of the modified incomplete Beta function. It is shown to have an application in the mathematics of multiple zeta functions , particularly, Euler sums. Example calculations are shown.
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.