$x\in \mathbb{R}\setminus\{0\}$ and $x + \frac{1}{x}$ is an integer. Prove $\forall n\in \mathbb{N}$ $x^n + \frac{1}{x^n}$ is also an integer.
41141
Proving that $\frac{\phi^{400}+1}{\phi^{200}}$ is an integer.
7802
Showing $\prod\limits_{i<j} \frac{x_i-x_j}{i-j}$ is an integer
48503
Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
156356
Proof that $x^n+\frac{1}{x^n}\in \mathbb Z$ by complete induction
131771
Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
20593
Show that if $a \neq b$ and a and b are positive then $\frac{a}{b}+\frac{b}{a}$ is never an integer
156646
If $x$ is rational then $\,x+\frac{1}x \,$ is an integer $\iff x = \pm 1$
119278
The number $\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$ is always an integer
150642
How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?
24469
Prove $ 2<(1+\frac{1}{n})^{n}$
145274
Prove by induction $\sum \frac {1}{2^n} < 1$
31245
How to prove that Fibonacci number is integer?
21746
How do I show that the sum $(a+\frac12)^n+(b+\frac12)^n$ is an integer for only finitely many $n$?
62792
Can sum of a rational number and its reciprocal be an integer?
102039
If any integer to the power of $x$ is integer, must $x$ be integer?
151153
Prove $\frac{a(a^2+2)}{3}$ is an integer for all integer $a\geqslant 1$
100080
Proving with induction $(1-x)^n<\frac 1 {1+nx}$