Intereting Posts

What are some good iPhone/iPod Touch/iPad Apps for mathematicians?
Uniform Continuity of $x \sin x$
Problem in Jacobson's Basic Algebra (Vol. I)
Ideals in $\mathbb{Z}$ with three generators (and not with two)
Showing ${1\over n}\sum|S_i|=O(\sqrt n)$ for $S_i\subset $, $|S_i\cap S_i|\le 1$ for $i\ne j$
Inequality concerning inverses of positive definite matrices
If $R=\{(x,y): x\text{ is wife of } y\}$, then is $R$ transitive?
How do you split a long exact sequence into short exact sequences?
Faithful functors from Rel, the category of sets and relations?
Prove that $\sum\limits_{j=k}^n\,(-1)^{j-k}\,\binom{j}{k}\,\binom{2n-j}{j}\,2^{2(n-j)}=\binom{2n+1}{2k+1}$.
Spaces with the property: Uniformly continuous equals continuous
How does $e^{\pi i}$ equal $-1$
If $f$ is a non-constant analytic function on $B$ such that $|f|$ is a constant on $\partial B$, then $f$ must have a zero in $B$
finding right quotient of languages
Has there ever been an application of dividing by zero?

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:

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

- The density — or otherwise — of $\{\{2^N\,\alpha\}:N\in\mathbb{N}\}$ for ALL irrational $\alpha$.
- Sum of rational numbers
- Is the difference of two irrationals which are each contained under a single square root irrational?
- Sine values being rational
- Irrational numbers to the power of other irrational numbers: A beautiful proof question
- Direct proof that $\pi$ is not constructible

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`

.

- Given $p,q$ odd primes, prove that if $\gcd(a,pq)=1$ then $a^{lcm (p-1,q-1)} \equiv 1 \pmod {pq}$
- Does there exist coprime numbers $a$ and $b$ such that $a^n+b$ is composite for every $n$?
- Does $\sum_{n=1}^\infty \frac{1}{p_ng_n}$ diverge?
- show that $\frac{a^2+b^2+c^2}{15}$ is non-square integer
- Is zero positive or negative?
- Show that $e^{\sqrt 2}$ is irrational
- Is $\lim_{k \to \infty}\left}\right]=1$?
- Finding all solutions of $x^{11}\equiv 1\bmod23,$
- Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?
- numbers' pattern

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.

- Let $f$ be a continuous function satisfying $\lim \limits_{n \to \infty}f(x+n) = \infty$ for all $x$. Does $f$ satisfy $f(x) \to \infty$?
- proof of $\sum_{n=1}^\infty n \cdot x^n= \frac{x}{(x-1)^2}$
- Under what conditions will the rectangle of the Japanese theorem be a square?
- Differentiating an infinite sum
- Prove that $ \left(a_{n}\right)_{n=1}^{\infty} $ converges when $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ for $ 0<q<1 $
- Comparing probabilities of drawing balls of certain color, with and without replacement
- If $n\in\mathbb N$ and $k\in\mathbb Z$, solve $n^3-32n^2+n=k^2$.
- Metrizability of weak convergence by the bounded Lipschitz metric
- Bivariate Normal Conditional Variance
- How to get intuition in topology concerning the definitions?
- The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?
- Infinite series of nth root of n factorial
- Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.
- Why is the eigenvector of a covariance matrix equal to a principal component?
- $U_q$ Quantum group and the four variables: $E, F, K, K^{-1}$