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).
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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
The lower right triangle has altitude relative to the upper-left corner, the perpendicular foot is obtained from the intersection point by multiplication with factor This gives an alternative tiling in aspect ratios with diagonal lengths in ratios and tiles of equal area opposite the diagonal.
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes from 1 and 2. Starting from 0 and 1, the sequence begins
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
The logistic map is a polynomial mapping of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations. The map, initially utilized by Edward Lorenz in the 1960s to showcase irregular solutions, was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst. Mathematically, the logistic map is written
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y.
In mathematics, the error function, often denoted by erf, is a function defined as:
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The square root of 2 is a positive real number that, when multiplied by itself or squared, equals the number 2. It may be written in mathematics as or . It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.
The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
In mathematics, a square triangular number is a number which is both a triangular number and a square number. There are infinitely many square triangular numbers; the first few are:
In mathematics, two quantities are in the silver ratio if the ratio of the larger of those two quantities to the smaller quantity is the same as the ratio of the sum of the smaller quantity plus twice the larger quantity to the larger quantity. This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.
In mathematics, the plastic ratio is a geometrical proportion close to 53/40. Its true value is the real solution of the equation x3 = x + 1.
In mathematics, the Fibonacci numbers form a sequence defined recursively by:
The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:
In mathematics, the Schröder number also called a large Schröder number or big Schröder number, describes the number of lattice paths from the southwest corner of an grid to the northeast corner using only single steps north, northeast, or east, that do not rise above the SW–NE diagonal.
In mathematics, the Perrin numbers are a doubly infinite constant-recursive integer sequence with characteristic equation x3 = x + 1. The Perrin numbers bear the same relationship to the Padovan sequence as the Lucas numbers do to the Fibonacci sequence.
The metallic mean of a natural number n is a positive real number, denoted here that satisfies the following equivalent characterizations:
In mathematics, an infinite sequence of numbers is called constant-recursive if it satisfies an equation of the form
In mathematics, the supergolden ratio is a geometrical proportion close to 85/58. Its true value is the real solution of the equation x3 = x2 + 1.
The square root of 7 is the positive real number that, when multiplied by itself, gives the prime number 7. It is more precisely called the principal square root of 7, to distinguish it from the negative number with the same property. This number appears in various geometric and number-theoretic contexts. It can be denoted in surd form as:
Table of Hilbert class polynomials