query: Prove that there do not exist nonzero integers $a$ and $b$ such that $a^2=3b^2$.
positive_passages.docid: 6275
positive_passages.text: How can you prove that the square root of two is irrational?
negative_passages.docid: 125164
negative_passages.text: How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
negative_passages.docid: 23301
negative_passages.text: There does not exist any integer $m$ such that $3n^2+3n+7=m^3$
negative_passages.docid: 63182
negative_passages.text: For every $n \ge 1$ there exist uniquely determined integers $a \gt 0$ and $b \gt 0$ such that $n = a^2b$ where $b$ is square-free.
negative_passages.docid: 156989
negative_passages.text: Proof that Hadamard matrices of order $4k+2$ don't exist
negative_passages.docid: 76032
negative_passages.text: Prove that if m is a square integer then m is neither congruent to 2 modulo 5 nor congruent to 3 modulo 5
negative_passages.docid: 89677
negative_passages.text: Why there isn't any integer solutions in non-zero integers for $z^3 = 3(x^3 +y^3+2xyz)$?
negative_passages.docid: 26764
negative_passages.text: Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)
negative_passages.docid: 127566
negative_passages.text: $2AB$ is a perfect square and $A+B$ is not a perfect square
negative_passages.docid: 74229
negative_passages.text: $2x^2+3y^2=z^2$ does not have a non-zero integral solution.
negative_passages.docid: 71110
negative_passages.text: Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$
negative_passages.docid: 135750
negative_passages.text: Prove that if a and b are integers, then there are unique integers q and r such that $a = bq + r$, $-|b|/2 < r \le |b|/2$
negative_passages.docid: 59653
negative_passages.text: $x^2-y^2=2s$, s cannot be an odd integer
negative_passages.docid: 23523
negative_passages.text: Two problems on number theory
negative_passages.docid: 125043
negative_passages.text: Show that F(A−B) is not necessarily a subset of F(A)−F(B).
negative_passages.docid: 102154
negative_passages.text: If $a$ and $b$ are consecutive integers, prove that $a^2 + b^2 + a^2b^2$ is a perfect square.
negative_passages.docid: 139841
negative_passages.text: Proving $A\setminus(B\setminus C)$ is not equal to $(A\setminus B)\setminus C$