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I recently came across a post on SO, asking to calculate the least two decimal digits of the integer part of $(4+\sqrt{11})^{n}$, for any integer $n \geq 2$ (see here).

The author presented a Java implementation using a BigInteger class and so forth, but the answer (given by someone else) was much simpler:

$$\forall n \geq 2 : \lfloor(4+\sqrt{11})^{n}\rfloor \pmod{ 100}=\lfloor(4+\sqrt{11})^{n+20}\rfloor \pmod {100}.$$

So in essence, we only need to calculate $(4+\sqrt{11})^{n}$ for every $n$ between $2$ and $21$.

I have unsuccessfully attempted to find a counterexample, using a BigRational class and a fast-converging $n$th-root algorithm for calculating $\sqrt[n]{A}$.

My question is then, how can we prove or refute the conjecture above?

Pardon my tags on this question, wasn't sure what else to put besides irrational-numbers.

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    $\begingroup$ Could you perhaps link to the SO post? $\endgroup$
    – quid
    Commented Feb 23, 2014 at 15:45
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    $\begingroup$ This is probably more appropriate for m.SE, but it is an interesting question! $\endgroup$
    – Noah Snyder
    Commented Feb 23, 2014 at 15:58
  • $\begingroup$ @quid, done; thanks. $\endgroup$
    – barak manos
    Commented Feb 23, 2014 at 16:00

1 Answer 1

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Consider the linear recurrence relation $a_{n+2} = 8a_{n+1} - 5a_n$ with $a_0 = 2$ and $a_1 = 8$. It is clear that $a_n$ is an integer for all $n \geq 0$. Also, this recurrence has the following closed-form solution for all $n \geq 0$:

$a_n = (4+ \sqrt{11})^n + (4 - \sqrt{11})^n$

Since $0 < 4 - \sqrt{11} < 1$, it follows that $\lfloor(4+ \sqrt{11})^n\rfloor = a_n - 1$ for all $n \geq 0$. Periodicity of $\lfloor(4+ \sqrt{11})^n\rfloor$ follows almost immediately now: first we check that $54 = a_{2} \equiv a_{22} \text{ mod } 100$ [Edit: and $92 \equiv a_{3} \equiv a_{23} \text{ mod } 100$]. Since $a_n$ is defined via a linear recurrence, it follows that $a_n \equiv a_{n+20} \text{ mod } 100$ for all $n \geq 2$, so $\lfloor(4+ \sqrt{11})^n\rfloor \equiv \lfloor(4+ \sqrt{11})^{n+20}\rfloor \text{ mod } 100$ for all $n \geq 2$ as well.

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    $\begingroup$ @barak -- Sure. I wanted the polynomial whose roots are $(4 + \sqrt{11})$ and $(4 - \sqrt{11})$, which turns out to be $x^2 - 8x + 5$, which corresponds to the linear recurrence I wrote. The initial conditions $a_0 = 2$ and $a_1 = 8$ were just chosen to make the answer "clean" -- they don't really matter. $\endgroup$
    – Nathaniel Johnston
    Commented Feb 23, 2014 at 16:12
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    $\begingroup$ Nice answer. Just, perhaps I am confused, but why don't you need to check also the congruence for a second value? $\endgroup$
    – quid
    Commented Feb 23, 2014 at 16:15
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    $\begingroup$ @quid -- Whoops. You're not confused. You do also have to check that $a_3 \equiv a_{23} \text{ mod } 100$ (fortunately that's true). $\endgroup$
    – Nathaniel Johnston
    Commented Feb 23, 2014 at 16:17
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    $\begingroup$ @barakmanos: It may help you to first read an explanation of why the Fibonacci numbers have the closed form $\frac{1}{\sqrt{5}} \left(\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^{n}\right)$. There's a general process for going getting closed forms of linear recurrences the same way, and Nathaniel just ran that process backwards starting with $(4+\sqrt{11})^n + (4-\sqrt{11})^n$. $\endgroup$
    – Noah Snyder
    Commented Feb 23, 2014 at 16:18
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    $\begingroup$ @T.J. Once you know $a_2 \equiv a_{22}$ and $a_3 \equiv a_{23}$ the recurrence tells you $a_4 = 8a_3-5a_2 \equiv 8a_{23}-5a_{22} = a_{24}$. Now just do that over and over again. $\endgroup$
    – Noah Snyder
    Commented Feb 24, 2014 at 14:29

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