query: Prove that $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ can not all be terms of a single arithmetic progression
positive_passages.docid: 148593
positive_passages.text: A sequence includes $a_p=\sqrt2$, $a_q=\sqrt3$, $a_r=\sqrt5$ for some $1\leq p<q<r$. Can these be terms of an arithmetic progression? harmonic?
negative_passages.docid: 23830
negative_passages.text: Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares
negative_passages.docid: 113986
negative_passages.text: Fermat's Last Theorem: implications (there is no new proof)
negative_passages.docid: 106716
negative_passages.text: The sum of square roots of non-perfect squares is never integer
negative_passages.docid: 146546
negative_passages.text: Proving for all integer $n \ge 2$, $\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$
negative_passages.docid: 67605
negative_passages.text: Different proof that $\sqrt{2}$ is irrational
negative_passages.docid: 114793
negative_passages.text: Prove that $\sqrt{10} - \sqrt6 - \sqrt5 + \sqrt3$ is irrational
negative_passages.docid: 145558
negative_passages.text: Prove that $2^n+3^n $ is never a perfect square
negative_passages.docid: 109953
negative_passages.text: How to prove that $\sqrt 3$ is an irrational number?
negative_passages.docid: 156032
negative_passages.text: Find a such that $112_a$, $63_a$ and $34_a$ are in an arithmetic progression
negative_passages.docid: 115976
negative_passages.text: $\mathbf{Q}[\sqrt 5+\sqrt[3] 2]=\mathbf{Q}[\sqrt 5,\sqrt[3] 2]$?
negative_passages.docid: 29874
negative_passages.text: Not divisible by $2,3$ or $5$ but divisible by $7$
negative_passages.docid: 109305
negative_passages.text: Non-existence of natural numbers such that $\sqrt{n} +\sqrt{n+1} <\sqrt{x} +\sqrt{y} <\sqrt{4n+2}$
negative_passages.docid: 56802
negative_passages.text: Why $\sqrt[3]{{2 + \sqrt 5 }} + \sqrt[3]{{2 - \sqrt 5 }}$ is a rational number?
negative_passages.docid: 37219
negative_passages.text: Quadratic Integers in $\mathbb Q[\sqrt{-5}]$
negative_passages.docid: 44440
negative_passages.text: Prove $\sqrt{2}+\sqrt[3]{5}$ is irrational
negative_passages.docid: 26857
negative_passages.text: In any Pythagorean triplet at least one of them is divisible by $2$, $3$ and $5$.