If $p$ is prime and any integer $k>1$, then $p^{\frac 1k} $ is irrational. Prove this by assuming $p^{\frac 1k}$ rational
44600
How can I prove: if $p $ is prime and $n>1$, then $ p^{\frac1n} $ is irrational?
111037
Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime.
18360
Proof that the irrational numbers are uncountable
111094
Help with proving that $\pi$ is irrational
95111
Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
119657
If $ f _p(x)=1+x+x^2+ \cdots +x^{p-1} $ where $p$ is prime number, $e$ is any positive integer.
5529
Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$
59391
$p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.
67605
Different proof that $\sqrt{2}$ is irrational
31867
Elementary proof that the limit of $\sum_{i=1}^{\infty} \frac{1}{\operatorname{lcm}(1,2,...,i)}$ is irrational
104548
Geometric proof for irrationality of $\pi$
126199
Proof of $\pi$ not being a quadratic irrational number.
158730
Is $\ln(\ln(n))$ irrational for any integer $n>1$?
81150
Positive integer multiples of an irrational mod 1 are dense
1383
How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
153862
A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.
127830
proof that $\sqrt(2)$ is irrational / why is the fraction irreducible