Plastic number
| Binary | 1.01010011001000001011… |
| Decimal | 1.32471795724474602596… |
| Hexadecimal | 1.5320B74ECA44ADAC1788… |
| Continued fraction |
[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80 ...] Note that this continued fraction is neither finite nor periodic. (Shown in linear notation) |
| Algebraic form | |
![\frac{\sqrt[3]{108 + 12\sqrt{69}} + \sqrt[3]{108 - 12\sqrt{69}}}{6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8abac975536b5091c5719656545cb7228cc4d4b)
In mathematics, the plastic number ρ (also known as the plastic constant or the minimal Pisot number) is a mathematical constant which is the unique real solution of the cubic equation
It has the exact value
Its decimal expansion begins with 1.324717957244746025960908854…. and at least 10,000,000,000 decimal digits have been computed.
The plastic number is also sometimes called the silver number, but that name is more commonly used for the silver ratio 1 + √2.
The powers of the plastic number A(n) = ρn satisfy the recurrence relation A(n) = A(n − 2) + A(n − 3) for n > 2. Hence it is the limiting ratio of successive terms of any (non-zero) integer sequence satisfying this recurrence such as the Padovan sequence and the Perrin sequence, and bears the same relationship to these sequences as the golden ratio does to the Fibonacci sequence and the silver ratio does to the Pell numbers.
The plastic number satisfies the nested radical recurrence:
Because the plastic number has minimal polynomial x3 − x − 1 = 0, it is also a solution of the polynomial equation p(x) = 0 for every polynomial p that is a multiple of x3 − x − 1, but not for any other polynomials with integer coefficients. Since the discriminant of its minimal polynomial is −23, its splitting field over rationals is ℚ(√−23, ρ). This field is also Hilbert class field of ℚ(√−23).
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