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.

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.

Thought on phi, the Golden Ratio.

I once argued (in “Shtetl Optimized”) that phi turns up a lot because it is one of the 20 simplest (complex) algebraic numbers, as follows:

Assign a rank(x) to x as rank(x) = deg(x) + coefs(x), where

deg(x) = minimal degree as an algebraic number: 0 for integers, 1 for rational, 2 for square roots, etc.

coefs(x) = absolute sum of integer coefficients in minimal polynomial for x.

rank(x) = deg(x) + coefs(x)

In order of rank, with pm = plus or minus, and the total at the end:

rank = 2: 1x + 0 = 0; (1)

rank = 3: 1x + pm 1 = 0; (2)

rank = 4: 1x + pm 2 = 0; 2x +pm 1 = 0; 1x^2 + 0x +1 = 0 (5)

rank = 5: 1x + pm 3 = 0, 1x^2 + 0x + pm2 = 0; 3x + pm1 = 0, 1x^2 + 0x +pm 2 = 0, 1x^2 +pm 1x +pm 1 = 0 (12)

The last equation: x^2 – x – 1 = 0 gives phi. with a total of 20 numbers through rank 5.

Do you think this ranking for algebraic numbers is a useful idea?

Ralph