# Thread: A near-integer expression

1. ## A near-integer expression

Computation shows that

$\displaystyle f(n) = \left( 1 + 2 \cos(\frac{\pi}{9}) \right) ^ n$

is, for "large n" (10 or greater, say), just a little less than an integer. For example,

$\displaystyle f(10) \approx 39173.98$
$\displaystyle f(20) \approx 1534601033.9998$
$\displaystyle f(30) \approx 60116436578864.999997$

Why is this? (I do not know the answer.)

A related thread is here:

http://www.mathhelpforum.com/math-he...r-divisib.html

2. If the integer part of x is denoted by [tex][x][/Math] fractional part of x is denoted by $\displaystyle \{x\}$, then you can see the following:

$\displaystyle \{f(20)\} = 0.9997998$

$\displaystyle \{f(21)\} = 0.9998732$

$\displaystyle \{f(22)\} = 0.9999152$

$\displaystyle \{f(23)\} = 0.9999458$

$\displaystyle \{f(24)\} = 0.9999640$

You can formulate it as a statement: f(n) gets arbitrarily close to an integer as n goes to infinity. Then maybe you can prove it. This is better than what you state, because we not only see that the function is close to an integer for big n, but it's getting closer and closer to an integer.

$\displaystyle \{f(100)\} = 0.9999999999999999997$

3. Here's an outline of a proof. No time for details because I'm about to go away for a week's holiday (again — that's the advantage of being retired).

The number $\displaystyle \alpha = 1+2\cos(2\pi/9)$ satisfies the equation $\displaystyle z^3-3z^2+1=0$ (see here). The other two roots of this equation, call them $\displaystyle \beta$ and $\displaystyle \gamma$, satisfy $\displaystyle |\beta|<1$ and $\displaystyle |\gamma}<1$.

Define a sequence of integers by $\displaystyle x_1=3,\;x_2=9,\;x_3=24$ and $\displaystyle x_n = 3x_{n-1}-x_{n-3}$ for $\displaystyle n>3$.

Then the theory of difference equations shows that $\displaystyle x_n = \alpha^n + \beta^n + \gamma^n$. Since $\displaystyle |\beta^n|\to0$ and $\displaystyle |\gamma^n|\to0$ as n gets large, it follows that $\displaystyle \alpha^n$ is only slightly different from an integer.

4. Wow, that's so neat! Is the following true?

Let the solutions to a polynomial equation be $\displaystyle \alpha_k$ for k from 1 to n. Then $\displaystyle \sum_{k=1}^{n}\alpha_k^n \in Z$.

EDIT: Never mind. I found some easy counterexamples.

5. I found something very interesting. (1+2cos(pi/9)) is what's called a PV number. Pisot . Check it out!

PV numbers are such numbers whose powers reach integers very fast, like the one we have here.