query: Positive integers $a$ and $b$ are such that $ab+1$ is a factor of $a^2+b^2$. Prove that $\frac{a^2+b^2}{ab+1}$ is a perfect square.
positive_passages.docid: 11506
positive_passages.text: Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer
negative_passages.docid: 134121
negative_passages.text: Determining all the positive integers $n$ such that $n^4+n^3+n^2+n+1$ is a perfect square.
negative_passages.docid: 134
negative_passages.text: Prove that the expression is a perfect square
negative_passages.docid: 151153
negative_passages.text: Prove $\frac{a(a^2+2)}{3}$ is an integer for all integer $a\geqslant 1$
negative_passages.docid: 19588
negative_passages.text: $n!+1$ being a perfect square
negative_passages.docid: 63383
negative_passages.text: Integers $n$ for which $|2^n + 5^n – 65|$ is a perfect square
negative_passages.docid: 90670
negative_passages.text: Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
negative_passages.docid: 64654
negative_passages.text: Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
negative_passages.docid: 100913
negative_passages.text: It is impossible for $(x-1)^2+x^2+(x+1)^2$ to be a perfect square
negative_passages.docid: 89329
negative_passages.text: If a and b are relatively prime and ab is a square, then a and b are squares.
negative_passages.docid: 66558
negative_passages.text: Find the value of $a$ and $b$ if $x+1$ and $x−2$ are factors of $ax^3−4x^2+bx−12$
negative_passages.docid: 120498
negative_passages.text: Why perfect square has odd number of factors
negative_passages.docid: 70308
negative_passages.text: Prove that if $p$ and $p+2$ are both prime integers greater than $3$, then $6$ is a factor of $p+1$.
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: 5692
negative_passages.text: Proving that there is a perfect square between $n$ and $2n$
negative_passages.docid: 138216
negative_passages.text: $a,b,c$ are positive real numbers such that, $a+b+c\ge abc$. Prove that $a^2+b^2+c^2\ge \sqrt{3}abc$
negative_passages.docid: 40626
negative_passages.text: Suppose that $ a,b \in \mathbb{Z}$ If 4 divides $ a^2 + b^2$, then $a$ and $b$ are not odd.