query: Number theory: find $a, b$ such that $\frac{a}{b} = b.a$ in a general base $\mathcal{B}\neq 10$
positive_passages.docid: 23315
positive_passages.text: Solve $\frac{b}{a}=a.b$ in decimal
negative_passages.docid: 52305
negative_passages.text: Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b,c > 0 $?
negative_passages.docid: 18693
negative_passages.text: If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
negative_passages.docid: 74631
negative_passages.text: For natural numbers $a$ and $b$, show that $a \Bbb Z + b \Bbb Z = \gcd(a, b)\Bbb Z $
negative_passages.docid: 76925
negative_passages.text: Algebraic proof that if $a>0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $
negative_passages.docid: 18772
negative_passages.text: If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.
negative_passages.docid: 61186
negative_passages.text: For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$
negative_passages.docid: 26287
negative_passages.text: Find $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ if $a+b+c=0$
negative_passages.docid: 1190
negative_passages.text: Square Root of Rational Number $\frac{A}{B}$
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: 6332
negative_passages.text: If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
negative_passages.docid: 96630
negative_passages.text: Let a and b be real numbers such that 0  \sqrt{ab} > \frac{2ab}{a + b}$
negative_passages.docid: 137967
negative_passages.text: Are there $a,b \in \mathbb{N}$ that ${(\sum_{k=1}^n k)}^a = \sum_{k=1}^n k^b $ beside $2,3$
negative_passages.docid: 58654
negative_passages.text: If $|A| > \frac{|G|}{2} $ then $AA = G $
negative_passages.docid: 98496
negative_passages.text: Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ iff $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$
negative_passages.docid: 135779
negative_passages.text: Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
negative_passages.docid: 49109
negative_passages.text: Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$