In mathematics, the supersilver ratio is a geometrical proportion close to 75/34. Its true value is the real solution of the equation x3 = 2x2 + 1.
The name supersilver ratio results from analogy with the silver ratio, the positive solution of the equation x2 = 2x + 1, and the supergolden ratio.
Two quantities a > b > 0 are in the supersilver ratio-squared if
The ratio is here denoted
Based on this definition, one has
It follows that the supersilver ratio is found as the unique real solution of the cubic equation The decimal expansion of the root begins as (sequence A356035 in the OEIS).
The minimal polynomial for the reciprocal root is the depressed cubic thus the simplest solution with Cardano's formula,
or, using the hyperbolic sine,
is the superstable fixed point of the iteration
Rewrite the minimal polynomial as , then the iteration results in the continued radical
Dividing the defining trinomial by one obtains , and the conjugate elements of are
with and
The growth rate of the average value of the n-th term of a random Fibonacci sequence is . [2]
The supersilver ratio can be expressed in terms of itself as the infinite geometric series
in comparison to the silver ratio identities
For every integer one has
Continued fraction pattern of a few low powers
The supersilver ratio is a Pisot number. [3] Because the absolute value of the algebraic conjugates is smaller than 1, powers of generate almost integers. For example: 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 has discriminant and factors into the imaginary quadratic field has class number Thus, the Hilbert class field of can be formed by adjoining [4] With argument a generator for the ring of integers of , the real root j(τ) of the Hilbert class polynomial is given by [5] [6]
The Weber-Ramanujan class invariant is approximated with error < 3.5 ∙ 10−20 by
while its true value is the single real root of the polynomial
The elliptic integral singular value [7] for has closed form expression
(which is less than 1/294 the eccentricity of the orbit of Venus).
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
with initial values
The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... (sequence A008998 in the OEIS). The limit ratio between consecutive terms is the supersilver ratio.
The first 8 indices n for which 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
The generating function of the sequence is given by
The third-order Pell numbers are related to sums of binomial coefficients by
The characteristic equation of the recurrence is If the three solutions are real root and conjugate pair and , the supersilver numbers can be computed with the Binet formula
Since and the number is the nearest integer to with n ≥ 0 and 0.1732702315504081807484794...
Coefficients result in the Binet formula for the related sequence
The first few terms are 3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008,... (sequence A332647 in the OEIS).
This third-order Pell-Lucas sequence has the Fermat property: if p is prime, The converse does not hold, but the small number of odd pseudoprimes makes the sequence special. The 14 odd composite numbers below 108 to pass the test are n = 32, 52, 53, 315, 99297, 222443, 418625, 9122185, 32572, 11889745, 20909625, 24299681, 64036831, 76917325. [10]
The third-order Pell numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue
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
and initiator . The series of words produced by iterating the substitution have the property that the number of c's, b's and a's are equal to successive third-order Pell numbers. The lengths of these words are given by [11]
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. [12]
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 1 and length . On the right-hand side, cut off a square of side length 1 and mark the intersection with the falling diagonal. The remaining rectangle now has aspect ratio (according to ). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point. [13]
Along the diagonal are two supersilver rectangles. The original rectangle and the scaled copies have diagonal lengths in the ratios or, equivalently, areas The areas of the rectangles opposite the diagonal are both equal to with aspect ratios (below) and (above).
The process can be repeated in the smallest supersilver rectangle at a scale of
Table of Hilbert class polynomials
In mathematics, the supersilver ratio is a geometrical proportion close to 75/34. Its true value is the real solution of the equation x3 = 2x2 + 1.
The name supersilver ratio results from analogy with the silver ratio, the positive solution of the equation x2 = 2x + 1, and the supergolden ratio.
Two quantities a > b > 0 are in the supersilver ratio-squared if
The ratio is here denoted
Based on this definition, one has
It follows that the supersilver ratio is found as the unique real solution of the cubic equation The decimal expansion of the root begins as (sequence A356035 in the OEIS).
The minimal polynomial for the reciprocal root is the depressed cubic thus the simplest solution with Cardano's formula,
or, using the hyperbolic sine,
is the superstable fixed point of the iteration
Rewrite the minimal polynomial as , then the iteration results in the continued radical
Dividing the defining trinomial by one obtains , and the conjugate elements of are
with and
The growth rate of the average value of the n-th term of a random Fibonacci sequence is . [2]
The supersilver ratio can be expressed in terms of itself as the infinite geometric series
in comparison to the silver ratio identities
For every integer one has
Continued fraction pattern of a few low powers
The supersilver ratio is a Pisot number. [3] Because the absolute value of the algebraic conjugates is smaller than 1, powers of generate almost integers. For example: 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 has discriminant and factors into the imaginary quadratic field has class number Thus, the Hilbert class field of can be formed by adjoining [4] With argument a generator for the ring of integers of , the real root j(τ) of the Hilbert class polynomial is given by [5] [6]
The Weber-Ramanujan class invariant is approximated with error < 3.5 ∙ 10−20 by
while its true value is the single real root of the polynomial
The elliptic integral singular value [7] for has closed form expression
(which is less than 1/294 the eccentricity of the orbit of Venus).
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
with initial values
The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... (sequence A008998 in the OEIS). The limit ratio between consecutive terms is the supersilver ratio.
The first 8 indices n for which 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
The generating function of the sequence is given by
The third-order Pell numbers are related to sums of binomial coefficients by
The characteristic equation of the recurrence is If the three solutions are real root and conjugate pair and , the supersilver numbers can be computed with the Binet formula
Since and the number is the nearest integer to with n ≥ 0 and 0.1732702315504081807484794...
Coefficients result in the Binet formula for the related sequence
The first few terms are 3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008,... (sequence A332647 in the OEIS).
This third-order Pell-Lucas sequence has the Fermat property: if p is prime, The converse does not hold, but the small number of odd pseudoprimes makes the sequence special. The 14 odd composite numbers below 108 to pass the test are n = 32, 52, 53, 315, 99297, 222443, 418625, 9122185, 32572, 11889745, 20909625, 24299681, 64036831, 76917325. [10]
The third-order Pell numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue
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
and initiator . The series of words produced by iterating the substitution have the property that the number of c's, b's and a's are equal to successive third-order Pell numbers. The lengths of these words are given by [11]
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. [12]
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 1 and length . On the right-hand side, cut off a square of side length 1 and mark the intersection with the falling diagonal. The remaining rectangle now has aspect ratio (according to ). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point. [13]
Along the diagonal are two supersilver rectangles. The original rectangle and the scaled copies have diagonal lengths in the ratios or, equivalently, areas The areas of the rectangles opposite the diagonal are both equal to with aspect ratios (below) and (above).
The process can be repeated in the smallest supersilver rectangle at a scale of
Table of Hilbert class polynomials