Plastic number

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The plastic number (also known as the plastic constant) is the unique real solution of the equation

x^3=x+1,\;

and has the value

\rho = \sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+\sqrt[3]{\frac{1}{2}-\frac{1}{6}\sqrt{\frac{23}{3}}}

which is approximately 1.324717957244746025960908854 (sequence A060006 in OEIS). The plastic number is also sometimes called the silver number, but that name is more commonly used for the silver ratio 1+\sqrt 2. The plastic number is the limiting ratio of successive terms of 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 is also a solution of the following equations:

x^5 = x^4 + 1\;
x^5 = x^2 + x + 1\;
x^6 = x^2 + 2x + 1\;
\!\ x^6 = x^4 + x + 1
x^7 = 2x^5 - 1\;
x^7 = 2x^4 + 1\;
x^8 = x^4 + x^3 + x^2 + x + 1\;
x^9 = x^6 + x^4 + x^2 + x + 1\;
x^{12} = 2x^{10} - x^4 - 1\;
x^{14} = 4x^9 + 1\;


The plastic number is the lowest Pisot-Vijayaraghavan number.

[edit] References

  • Midhat J. Gazalé, Gnomon, 1999 Princeton University Press.

[edit] External links

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