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
Perrin Sequences My Chalkboard 