New Insights into Integer Sequences
Language
English
Book Summary
What can be learned by studying integer sequences? As a chemist with an interest in mathematics I explore answering this question, finding it takes the reader into various disciplines such as linear algebra, physics, engineering, geometry, number theory and calculus. I begin by showing that a sequence is more than a series of numbers. Using the Perrin sequence which has been known for over 130 years, I first explore the relationship of prime numbers to finite sequence lengths of the Perrin sequence. Using programs like MathCad and Mathematica in later chapters, the student can verify any of the expressions and examples I introduce in each chapter. In each of the early Chapters I discuss new mathematical techniques needed for finding operations describing cubic and higher order polynomials. As a guide to understanding known sequences, I refer many times to the Online Encyclopedia of Integer Sequences (OEIS). By Chapter 11 the reader will have learned sufficient mathematics to understand concepts of partition functions, maximal independent sets, cycles and graphs and how they relate to the Perrin numbers. At this point in the book there is an emphasis on applications of integer sequences such as finding pseudoprimes, the distribution of prime numbers and new methods for calculating Perrin numbers. After a translation from French of Perrin’s brief Query in 1899, I review the development of number theory and combinatorics derived from Fibonacci, Padovan and Perrin sequences. As the golden ratio describes the Fibonacci series, the plastic number is used to build a geometry into the Perrin series. Various new linear recurrence relationships are then developed. The remainder of the book discusses orthogonal polynomials and modular functions and the development of new general equations using these polynomials to calculate sequence numbers of higher order Perrin polynomials of the form xg – Ax(g-j) -B = 0 where g and j are integers. The symmetry of many sequences is explored with these polynomials. Concepts of Ramanujan’s octave, Ramanujan’s ladder, the Complex Octagon and the solving of quintic equations with 19th century methods are chapters which culminate this book, providing the reader new insights into the arithmetic of algebraic numbers surrounding the integer sequence.
Richard S. Turk
July 2024 472pp
e-book is available
IN USA/North America: RSTURK@2RMTECHNOLOGY.COM or call/Text 1 5082694802
