Supersilver ratio explained

Rationality:	irrational algebraic
Continued Fraction Linear:	[2;4,1,6,2,1,1,1,1,1,1,2,2,1,2,1,...]
Continued Fraction Periodic:	not periodic
Continued Fraction Finite:	infinite
Algebraic:	real root of

In mathematics, the supersilver ratio is a geometrical proportion close to . Its true value is the real solution of the equation

The name supersilver ratio results from analogy with the silver ratio, the positive solution of the equation, and the supergolden ratio.

Definition

Two quantities are in the supersilver ratio-squared if

\left( (2 a + b)/(a) \right) ^{2} = (a)/(b) .The ratio (2 a + b)/(a) is here denoted

Based on this definition, one has

\begin{align} 1 & = \left( (2 a + b)/(a) \right) ^{2} (b)/(a) \\ & = \left( (2 a + b)/(a) \right) ^{2} \left( (2 a + b)/(a) - 2 \right)\\ &\implies \varsigma ^{2} \left( \varsigma - 2 \right) = 1 \end{align}

It follows that the supersilver ratio is found as the unique real solution of the cubic equation

\varsigma ^{3} -2 \varsigma ^{2} -1 =0 . The decimal expansion of the root begins as 2 . 205569430400590 ... .

The minimal polynomial for the reciprocal root is the depressed cubic

x ^{3} +2 x -1 , thus the simplest solution with Cardano's formula,w _{1,2} = \left( 1 \pm (1)/(3) \sqrt{ (59)/(3) } \right) /2 1 /\varsigma = \sqrt[ 3 ]{w _{1}} + \sqrt[ 3 ]{w _{2}} or, using the hyperbolic sine,1 /\varsigma =-2 \sqrt (2)/(3) \sinh \left( (1)/(3) \operatorname{arsinh} \left( - (3)/(\sqrt 4) (3)/(2) \right) \right).

is the superstable fixed point of the iteration

x \gets ( 2 x ^{3} +1 ) /( 3 x ^{2} +2 ).

Rewrite the minimal polynomial as

(x ^{2+1)} ^{2} =1+ x , then the iteration x \gets \sqrt{ -1 + \sqrt{ 1+ x}} results in the continued radical1 /\varsigma = \sqrt{ -1 + \sqrt{ 1 + \sqrt{ -1 + \sqrt{ 1 + … }}}}

Dividing the defining trinomial

x ^{3} -2 x ^{2} -1 by one obtains x ^{2} + x /\varsigma ^{2} +1 /\varsigma , and the conjugate elements of arex _{1,2} = \left( -1 \pm i \sqrt{ 8 \varsigma ^{2} +3 } \right) / 2 \varsigma ^{2,} with x _{1} + x _{2} = 2 - \varsigma and x _{1x} _{2} =1 /\varsigma.

Properties

The growth rate of the average value of the n-th term of a random Fibonacci sequence is .

The supersilver ratio can be expressed in terms of itself as the infinite geometric series

\varsigma = 2 \sum ^{inf ty} _{k =0} \varsigma ^{-3k} and \varsigma ^{2} =-1 + \sum ^{inf ty} _{k =0} (\varsigma -1 ) ^{-k} , in comparison to the silver ratio identities\sigma = 2 \sum ^{inf ty} _{k =0} \sigma ^{-2k} and \sigma ^{2} =-1 +2 \sum ^{inf ty} _{k =0} (\sigma -1 ) ^{-k} .

For every integer

n one has\begin{align} \varsigma ^{n} & =2 \varsigma ^{n-1} + \varsigma ^{n-3} \\ & =4 \varsigma ^{n-2} + \varsigma ^{n-3} +2 \varsigma ^{n-4} \\ & = \varsigma ^{n-1} +2 \varsigma ^{n-2} + \varsigma ^{n-3} + \varsigma ^{n-4} .\end{align}

Continued fraction pattern of a few low powers

\varsigma ^{-2} = [ 0 ; 4 , 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , 1 ,...] ≈ 0 . 2056 \varsigma ^{-1} = [ 0 ; 2 , 4 , 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 ,...] ≈ 0 . 4534 \varsigma ^{0} = [ 1 ] \varsigma ^{1} = [ 2 ; 4 , 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , 1 ,...] ≈ 2 . 2056 \varsigma ^{2} = [ 4 ; 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , 2 ,...] ≈ 4 . 8645 \varsigma ^{3} = [ 10 ; 1 , 2 , 1 , 2 , 4 , 4 , 2 , 2 , 6 , 2 ,...] ≈ 10 . 729 1 /\sqrt{\varsigma} of the algebraic conjugates is smaller than 1, powers of generate almost integers. For example: \varsigma ^{10} =2724 . 00146856 ... ≈ 2724 +1 / 681 . After ten rotation steps the phases of the inward spiraling conjugate pair  - initially close to  - nearly align with the imaginary axis.

The minimal polynomial of the supersilver ratio

m(x) = x ^{3} -2 x ^{2} -1 has discriminant \Delta =-59 and factors into (x -21 ) ^{2} (x -19 ) \pmod{ 59 }; the imaginary quadratic field K = Q ( \sqrt{\Delta}) has class number Thus, the Hilbert class field of can be formed by adjoining ^[2] With argument \tau = ( 1 + \sqrt{\Delta})/ 2 a generator for the ring of integers of, the real root of the Hilbert class polynomial is given by (\varsigma ^{-6} -27 \varsigma ^{6} -6 ) ^{3} . ^{[3] [4]}

The Weber-Ramanujan class invariant is approximated with error by

\sqrt{ 2 } ak{f}( \sqrt{ \Delta} ) = \sqrt[ 4 ]{ 2 }G _{59} ≈ (e ^{\pi} } + 24)^,while its true value is the single real root of the polynomialW _{59} (x) = x ^{9} -4 x ^{8} +4 x ^{7} -2 x ^{6} +4 x ^{5} -8 x ^{4} +4 x ^{3} -8 x ^{2} +16 x -8 . k _{r} =λ ^{*} (r) for has closed form expressionλ ^{*} ( 59 ) = \sin ( \arcsin \left( G ^{-12} _{59} \right) / 2 ) (which is less than 1/294 the eccentricity of the orbit of Venus).

Third-order Pell sequences

These numbers are related to the supersilver ratio as the Pell numbers and Pell-Lucas numbers are to the silver ratio.

The fundamental sequence is defined by the third-order recurrence relation

S _{n} =2 S _{n-1} + S _{n-3} for,with initial valuesS _{0} =1 , S _{1} =2 , S _{2} =4 .

The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... .The limit ratio between consecutive terms is the supersilver ratio.

The first 8 indices n for which

S _{n} is prime are n = 1, 6, 21, 114, 117, 849, 2418, 6144. The last number has 2111 decimal digits.

The sequence can be extended to negative indices using

S _{n} = S _{n+3} -2 S _{n+2} .

The generating function of the sequence is given by

(1)/(1-2 x - x ^{3}) = \sum ^{inf ty} _{n =0} S _{n} x ^{n} for x < 1 /\varsigma .

The third-order Pell numbers are related to sums of binomial coefficients by

S _{n} = \sum ^{\lfloor n / 3 \rfloor} _{k =0} {n -2 k \choose k} ⋅ 2 ^{n} .^[5]

The characteristic equation of the recurrence is

x ^{3} -2 x ^{2} -1 =0 . If the three solutions are real root and conjugate pair and, the supersilver numbers can be computed with the Binet formulaS _{n-2} = a \alpha ^{n} + b \beta ^{n} + c \gamma ^{n} , with real and conjugates and the roots of 59 x ^{3} +4 x -1 =0 . Since \left\vert b \beta ^{n} + c \gamma ^{n} \right\vert < 1 /\sqrt{ \alpha ^{n} } and \alpha = \varsigma, the number is the nearest integer to a\varsigma ^{n+2} , with and a = \varsigma /( 2 \varsigma ^{2} +3 ) =

Coefficients

a = b = c =1 result in the Binet formula for the related sequence A _{n} = S _{n} +2 S _{n-3} .

The first few terms are 3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008,... .

This third-order Pell-Lucas sequence has the Fermat property: if p is prime,

A _{p} \equiv A _{1} \bmod p. The converse does not hold, but the small number of odd pseudoprimes n \mid (A _{n} -2 ) makes the sequence special. The 14 odd composite numbers below to pass the test are n = 3, 5, 5, 315, 99297, 222443, 418625, 9122185, 3257, 11889745, 20909625, 24299681, 64036831, 76917325.^[6] Q = \begin{pmatrix} 2 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{pmatrix} ,

The trace of gives the above

Alternatively, can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet with corresponding substitution rule

\begin{cases} a \mapsto aab\\ b \mapsto c\\ c \mapsto a \end{cases} and initiator . The series of words produced by iterating the substitution have the property that the number of and are equal to successive third-order Pell numbers. The lengths of these words are given by l(w _{n)} = S _{n-2} + S _{n-3} + S _{n-4} . ^[7]

Associated to this string rewriting process is a compact set composed of self-similar tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation three-letter sequence.^[8]

Supersilver rectangle

A supersilver rectangle is a rectangle whose side lengths are in a ratio. Compared to the silver rectangle, containing a single scaled copy of itself, the supersilver rectangle has one more degree of self-similarity.

Given a rectangle of height and length . On the right-hand side, cut off a square of side length and mark the intersection with the falling diagonal. The remaining rectangle now has aspect ratio

1 +1 / \varsigma ^{2:1} (according to \varsigma =2 +1 / \varsigma ^{2} ). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.^[9]

Along the diagonal are two supersilver rectangles. The original rectangle and the scaled copies have diagonal lengths in the ratios

\varsigma /(\varsigma -1 ):(\varsigma -1 ): 1 or, equivalently, areas \varsigma ^{2/} (\varsigma -1 ) ^{2:(\varsigma} -1 ) ^{2:1.} The areas of the rectangles opposite the diagonal are both equal to (\varsigma -1 )/ \varsigma, with aspect ratios \varsigma +1 / \varsigma (below) and \varsigma /(\varsigma -1 ) (above).

The process can be repeated in the smallest supersilver rectangle at a scale of

1 :\varsigma.

The lower right triangle has altitude

1 /\sqrt{\varsigma -1 }; relative to the upper-left corner, the perpendicular foot is obtained from the intersection point \left( \varsigma -1 , (\varsigma -1)/(\varsigma) \right) by multiplication with factor 1 /(\varsigma +1 ). This gives an alternative tiling in aspect ratios \varsigma ^{2,\varsigma,1,\varsigma ,} with diagonal lengths in ratios (\varsigma -1 ) ^{2:\varsigma:1} and tiles of equal area opposite the diagonal.

See also

• Solutions of equations similar to

x ^{3} =2 x ^{2} +1 :

• • Silver ratio – the only positive solution of the equation

x ^{2} =2 x +1

• • Golden ratio – the only positive solution of the equation

x ^{2} = x +1

• • Supergolden ratio – the only real solution of the equation

x ^{3} = x ^{2} +1

Notes and References

1. Panju . Maysum . 2011 . A systematic construction of almost integers . The Waterloo Mathematics Review . 1 . 2 . 35–43.
2. Web site: Hilbert class field of a quadratic field whose class number is 3 . 2012 . Mathematics stack exchange . May 1, 2024.
3. Berndt . Bruce C.. Chan . Heng Huat . 1999 . Ramanujan and the modular j-invariant . . 42 . 4 . 427–440 . 10.4153/CMB-1999-050-1 . free.
4. Web site: Modular j-invariant . Johansson . Fredrik . 2021 . Fungrim . April 30, 2024 . Table of Hilbert class polynomials.
5. Mahon . Br. J. M. . Horadam . A. F. . 1990 . Third-order diagonal functions of Pell polynomials . . 28 . 1 . 3–10.
6. Only one of these is a 'restricted pseudoprime' as defined in: Adams . William . Shanks . Daniel . Daniel Shanks . Strong primality tests that are not sufficient . . . 1982 . 39 . 159 . 255–300 . 10.1090/S0025-5718-1982-0658231-9 . free. 2007637.
7. for n ≥ 2
8. Siegel . Anne . Thuswaldner . Jörg M. . 2009 . Topological properties of Rauzy fractals . Mémoires de la Société Mathématique de France . 118 . 2 . 1-140 . 10.24033/msmf.430.
9. Analogue to the construction in: Crilly . Tony . 1994 . A supergolden rectangle . . 78 . 483 . 320–325 . 10.2307/3620208.