Welcome to the Perrin Chalkboard! I will be presenting in this blog a series of chalkboards which discuss interesting properties of the Perrin sequence and related integer sequences. This blog starts as a simple discussion of the Perrin sequence (the original mention by Lucas in 1876 and Perrin in 1899). It is found that an immense amount of research on the associated elliptic curves has occurred over the last 115 years. The Perrin sequence ties together much of the mathematics discussed today as algebraic number theory and modular functions. It is also integral to the discussion of Fermat’s Last Theorem conjectured in 1637 but proved by Wiles in 1994.
Theorems will be presented without proofs. I think the subject matter will appeal to those interested in the properties of integer sequences, elliptic equations, and graph theory. Many sequences from OEIS (On-line Encyclopedia of Integer Sequences) will be discussed, uncovering hidden or less obvious properties.
The primary subject matter in this blog covers the properties of integer sequences. However, it is not until Chapter 17 that I cover the subject matter of the short paper published by Perrin in 1899. This chapter then introduces the subject of integer partitions, followed by some geometric applications of the Perrin sequence and then turns to division algorithms derived from general properties of cubic equations and associated integer sequences.
I encourage any comments or suggestions to the chalkboard subjects.
Although the pdf files are freely provided, if you are interested or have questions regarding any chapter please feel free to contact me.
Updated July 2017
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Discussion of further investigation of the connection between the octic q fraction, class invariants, tonality and the octahedron.
The Ramanujan Octave, Semitones, Chords and Harmonics_
The calculation of the g invariant for discriminants -4m requires an adjustment to the Ramanujan octave invariant for odd discriminants. The new invariant is shown to be useful for finding solvable polynomials if the q octic continued fraction is divided by harmonics of the 12-tone scale. The 1/12 and 1/8 powers of 12 are significant in finding monic polynomials of degree 4 or less. The numbers 8 and 12 also appear in the elements of the octahedron. This platonic solid has 8 faces, 12 edges and 6 vertices. The solid may have a natural relation to the 12-tone scale only based on the number 12 but it also extends into the mathematics of modular functions. The powers 1/12 and 1/24 are found in many modular equations such as the j-invariant where is a factor used to impose independence of the invariant for elliptic curves in any coordinate system. The powers 12, 24 and 8 appear in the Weber function relations to the j-invariant. Although other variants of the Ramanujan octave have appeared in the literature, the results discussed in this paper do not have any significance beyond these mathematical observations but still serves as an intriguing connection of mathematics to architectural geometry and music harmony.
The Ramanujan Octave and Discriminants -4m
An analogy is made between the equations derived Chapter 29 with the musical equal tempered scale. I call this the Ramanujan octave based on his octic q continued fraction and its relation to class invariants. The existence theorem of solvability is demonstrated with two examples representing the product and quotient of the modulus of the octic continued fraction. These examples show a universality to finding radical forms of the class invariants for any class number. An infinite number of semitones are shown to create the Ramanujan octave!
The Ramanujan Octave and Examples of the Existence Theorem
Following the P-Q modular equations used by B. Berndt (Transactions of the American Mathematical Society, 349(6), June 1997) to determine 13 radical forms of class invariants reported by Ramanujan, this Chapter discusses a new method of analysis. By using the q octic continued fraction and the q cubic solution equation (qkQ) derived in Chapter 28, all 13 radical forms are solved using results from Berndt’s two theorems. The radical forms for discriminants with prime divisors of 5 and 7 are presented without the need for modular P-Q equations.
A theorem of existence of a solution of the class invariant in radicals is presented based on an invariant of the class invariant and modulus of the octic q continued fraction.
expressing the octic q continued fraction in radical form using a q cubic solution equation
This appendix illustrates a calculation of one of Weber’s class invariants and compares the result obtained by Weber with the analysis presented in Chapter 28. The analytical method provides some extra parameters for finding a suitable irreducible polynomial that can be expressed in radical form. Consequently, the modulus of the q continued fraction is also expressed in radicals and/or nested radicals.
calculating a class invariant from ramanujan’s octic q continued fraction
This board is an extension of Chapter 20 on discriminants and modular functions. It provides a method of expressing the Rogers-Ramanujan octic q continued fraction in terms of radicals. Some interesting relations of the q continued fraction with the plastic number are shown. A q modulus equation is also derived to find new radical expressions for polynomials with various complex quadratic fields. Radical form is possible if the order of the associated polynomial with discrimination (-d) is less than five or provided the real root U of f(x) is solvable by radicals. Unexpectedly the later is true for all but four of the discriminants 1 or 3 mod 4 less than 100 using a radical extension of the irreducible coefficient field.
expressing a ramanujan q continued fraction in terms of radicals_
A look back at Chapter 23 reveals a new way to view the orthogonal Jacobi Polynomial. Through a series of identities it is possible to reduce any Jacobi polynomial P[m,a,b,x], with b> 1 and evaluated at x = 3 to a series of Jacobi polynomials with b = 1 or b = 0. The scalar vector product with the diagonal of the Chebychev T polynomial is introduced. It is shown that any Jacobi Polynomial P[m,a,b,3] is represented as vector products of an associated (m-1) row of the Pascal triangle and a shifted cycle index of the Symmetry group S(m). A combinatoric role of the Jacobi is suggested in this analysis.
The Jacobi polynomial can also be used to calculate the nth term of the Perrin sequence and the sigma orbit of prime numbers as described in Chapter 13.
The Jacobi Polynomial Revisited