query: Proving there is no rational number such that square of it is equal to 2.
positive_passages.docid: 143733
positive_passages.text: Is this proof that $\sqrt 2$ is irrational correct?
negative_passages.docid: 151439
negative_passages.text: $\sqrt{1-x^2}$ is not a rational function
negative_passages.docid: 86893
negative_passages.text: Show that it is impossible to list the rational numbers in increasing order
negative_passages.docid: 6582
negative_passages.text: Prove the existence of the square root of $2$.
negative_passages.docid: 135858
negative_passages.text: Prove $x^5 + x^4 + x^3 + x^2 + 1 = 0$ has no rational solution
negative_passages.docid: 148585
negative_passages.text: Show that, for any $\epsilon>0$, there exist two rationals such that $q < x < q'$ and $|q-q'|<\epsilon$
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: 45911
negative_passages.text: Prove that none of $\{11, 111, 1111,\dots \}$ is the perfect square of an integer
negative_passages.docid: 97114
negative_passages.text: Square is not an algebraic set.
negative_passages.docid: 78287
negative_passages.text: Prove that if $n$ is not the square of a natural number, then $\sqrt{n}$ is irrational.
negative_passages.docid: 39021
negative_passages.text: Proving that $\sqrt{2}+\sqrt{3}$ is irrational
negative_passages.docid: 67031
negative_passages.text: use contradiction to prove that the square root of $p$ is irrational
negative_passages.docid: 59653
negative_passages.text: $x^2-y^2=2s$, s cannot be an odd integer
negative_passages.docid: 52960
negative_passages.text: Show that the curve $x^2+y^2-3=0$ has no rational points
negative_passages.docid: 126494
negative_passages.text: Prove that there doesn't exist any integer $x \ge 3$ such that $x^2-1$ is prime.
negative_passages.docid: 136366
negative_passages.text: Show that if $2+2\sqrt{28n^2+1}$ is an integer then it must be perfect square.
negative_passages.docid: 96647
negative_passages.text: Number theory: Problem involving rational numbers