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.

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

The minimal polynomial for the reciprocal root is the depressed cubic

results in the continued radical Dividing the defining trinomial

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 fractions

Continued fraction pattern of a few low powers

After ten rotation steps the phases of the inward spiraling conjugate pair – initially close to ⁠

⁠, 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]

has closed form expression (which is less than 1/294 the eccentricity of the orbit of Venus).

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

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

⁠, the supersilver numbers can be computed with the Binet formula Since

result in the Binet formula for the related sequence

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

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 ⁠

⁠ can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet ⁠

⁠ 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.

[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.

each perpendicular foot divides the diagonal in ratio ⁠

Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.

[13] The parent supersilver rectangle and the two scaled copies along the diagonal have linear sizes in the ratios

The areas of the rectangles opposite the diagonal are both equal to

If the diagram is further subdivided by perpendicular lines through the feet of the altitudes, the lengths of the diagonal and its seven distinct subsections are in ratios

If drawn on a supersilver rectangle, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of rectangles with aspect ratio

which are perpendicularly aligned and successively scaled by a factor