In mathematics, the supersilver ratio is a geometrical proportion, given by the unique real solution of the equation x³ = 2x² + 1. Its decimal expansion begins with 2.2055694304005903... (sequence A356035 in the OEIS).

The name supersilver ratio is by analogy with the silver ratio, the positive solution of the equation x² = 2x + 1, and the supergolden ratio.

Three quantities a > b > c > 0 are in the supersilver ratio if $${\displaystyle {\frac {2a+c}{a}}={\frac {a}{b}}={\frac {b}{c}}\,.}$$ This ratio is commonly denoted ⁠${\displaystyle \varsigma }$⁠.

Substituting ${\displaystyle a=\varsigma b=\varsigma ^{2}c}$ in the first fraction gives $${\displaystyle \varsigma ={\frac {2\varsigma ^{2}c+c}{\varsigma ^{2}c}}.}$$ It follows that the supersilver ratio is the unique real solution of the cubic equation ${\displaystyle \varsigma ^{3}-2\varsigma ^{2}-1=0.}$

The minimal polynomial for the reciprocal root is the depressed cubic ${\displaystyle x^{3}+2x-1,}$ thus the simplest solution with Cardano's formula, $${\displaystyle {\begin{aligned}w_{1,2}&=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {59}{3}}}\right)/2\\1/\varsigma &={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}\end{aligned}}}$$ or, using the hyperbolic sine, $${\displaystyle 1/\varsigma =-2{\sqrt {\frac {2}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left(-{\frac {3}{4}}{\sqrt {\frac {3}{2}}}\right)\right).}$$

⁠${\displaystyle 1/\varsigma }$⁠ is the superstable fixed point of the iteration ${\displaystyle x\gets (2x^{3}+1)/(3x^{2}+2).}$

Rewrite the minimal polynomial as ${\displaystyle (x^{2}+1)^{2}=1+x}$ (multiplied by an additional factor of ${\displaystyle x}$, which harmlessly adds an additional root of 0); then the iteration ${\displaystyle x\gets {\sqrt {-1+{\sqrt {1+x}}}}}$ results in the continued radical  $${\displaystyle 1/\varsigma ={\sqrt {-1+{\sqrt {1+{\sqrt {-1+{\sqrt {1+\cdots }}}}}}}}}$$

Dividing the defining trinomial ${\displaystyle x^{3}-2x^{2}-1}$ by ⁠${\displaystyle x-\varsigma }$⁠ one obtains ${\displaystyle x^{2}+x/\varsigma ^{2}+1/\varsigma ,}$ and the conjugate elements of ⁠${\displaystyle \varsigma }$⁠ are $${\displaystyle x_{1,2}=\left(-1\pm i{\sqrt {8\varsigma ^{2}+3}}\right)/2\varsigma ^{2},}$$ with ${\displaystyle x_{1}+x_{2}=2-\varsigma \;}$ and ${\displaystyle \;x_{1}x_{2}=1/\varsigma .}$