$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational
24301
Can a finite sum of square roots be an integer?
53945
The square root of a prime number is irrational
32848
Prove that the product of a non-zero rational and irrational number is irrational.
21457
Prove $\sqrt{2\sqrt{3\sqrt{\cdots\sqrt{n}}}}<3$ by induction.
127830
proof that $\sqrt(2)$ is irrational / why is the fraction irreducible
23752
Proving Irrationality
19676
Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?
124641
Prove that $5^{1/3}+7^{1/2}$ is irrational
143733
Is this proof that $\sqrt 2$ is irrational correct?
5927
How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$
58284
How to prove that $\sqrt[3] 2 + \sqrt[3] 4$ is irrational?
5529
Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$
26109
Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$
34349
Prove that $\frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n} \geq \frac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$
90276
Is $\sqrt{2}\in{\Bbb Z}[\sqrt{2}+\sqrt{3}]$ true?
147777
If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?
59391
$p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.