Last week I wrote a blog post showing that powers of the golden ratio are nearly integers. Specifically, the distance from φ^{n} to the nearest integer decreases exponentially as *n* increases. Several people pointed out that the golden constant is a Pisot number, the general class of numbers whose powers are exponentially close to integers.

The so-called plastic constant *P* is another Pisot number, in fact the smallest Pisot number. *P* is the real root of *x*^{3} – *x* – 1 = 0.

Because *P* is a Pisot number, we know that its powers will be close to integers, just like powers of the golden ratio, but the *way* they approach integers is more interesting. The convergence is slower and less regular.

We will the first few powers of *P*, first looking at the distance to the nearest integer on a linear scale, then looking at the absolute value of the distance on a logarithmic scale.

As a reminder, here’s what the corresponding plots looked like for the golden ratio.

Another good post, thanks!

One significant typo — in the math markup, the numerator has “9 minus root 69” twice; it should have “minus” once and “plus” once.

Thanks. Just fixed the typo.

Every nth Lucas number (2, 1, 3, 4, 7, …) is a sum of nth power of golden ratio and its negative nth power multiplied by (-1) to the nth. L(n) = Phi^n + phi^ (-n) . For instance: L(3) = 4 and adequately Phi^3 + phi^(-3) = 4. Consequently, with growing n value the second component the above sum vanishes and the first term asymptotically approaches the adequate Lucas number.

Regards,

PG13

BTW, the ratio of subsequent Lucas numbers L(n)/L(n-1) converges to the Golden Ratio.