query: If $2xy$ divides $x^2+y^2-x$, prove that $x$ is a perfect square
positive_passages.docid: 142513
positive_passages.text: Prove that $2xy\mid x^2+y^2-x$ implies $x$ is a perfect square.
negative_passages.docid: 147666
negative_passages.text: How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?
negative_passages.docid: 155789
negative_passages.text: Integral values of $x$ for which the expression $x^2+19x+92$ is a perfect square
negative_passages.docid: 67968
negative_passages.text: When is $2^n -7$ a perfect square?
negative_passages.docid: 122115
negative_passages.text: Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
negative_passages.docid: 154082
negative_passages.text: Product of factorials divided by factorial to produce perfect square
negative_passages.docid: 1448
negative_passages.text: $x^a - 1$ divides $x^b - 1$ if and only if $a$ divides $b$
negative_passages.docid: 57368
negative_passages.text: Prove that a positive polynomial function can be written as the squares of two polynomial functions
negative_passages.docid: 36906
negative_passages.text: Prove that $x^n+x^{-n} \in \mathbf{N}$ if $x+\frac1x \in \mathbf{N}$
negative_passages.docid: 155784
negative_passages.text: If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers
negative_passages.docid: 54355
negative_passages.text: $1+x+\ldots+x^n$ perfect square
negative_passages.docid: 10548
negative_passages.text: The equation $-1 = x^2 + y^2$ in finite fields
negative_passages.docid: 80144
negative_passages.text: Prove that $x-1$ divides $x^n-1$
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: 125267
negative_passages.text: Prove that $2^{1/2}$ is irrational
negative_passages.docid: 91877
negative_passages.text: Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
negative_passages.docid: 78593
negative_passages.text: Prove that the product of four consecutive positive integers plus one is a perfect square