query: Show that if $x,y,z$ are positive integers, then $(xy + 1)(yz + 1)(zx + 1)$ is a perfect square iff $xy +1, yz +1, zx+1$ are all perfect squares.
positive_passages.docid: 152643
positive_passages.text: $(xy + 1)(yz + 1)(zx + 1) $ is a perfect square if and only if each factor is
negative_passages.docid: 63383
negative_passages.text: Integers $n$ for which $|2^n + 5^n – 65|$ is a perfect square
negative_passages.docid: 18313
negative_passages.text: show this inequality with $xy+yz+zx=3$
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: 137626
negative_passages.text: If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
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: 54355
negative_passages.text: $1+x+\ldots+x^n$ perfect square
negative_passages.docid: 24446
negative_passages.text: Prove that for each $n$, there exist $n$ consecutive integers, each of which is divisible by a perfect square larger than $1$.
negative_passages.docid: 116223
negative_passages.text: Show that the angles satisfy $x+y=z$
negative_passages.docid: 135212
negative_passages.text: The square of an integer has form $3n$ or $3n+1$
negative_passages.docid: 80671
negative_passages.text: Product of sums of square is a sum of squares.
negative_passages.docid: 155757
negative_passages.text: Number theory: $x^y + 1 = y^x$
negative_passages.docid: 130499
negative_passages.text: How can $p^{q+1}+q^{p+1}$ be a perfect square?
negative_passages.docid: 127566
negative_passages.text: $2AB$ is a perfect square and $A+B$ is not a perfect square
negative_passages.docid: 152167
negative_passages.text: Prove that if $x,y,z$ are integers such that $x^2+y^2=z^2$, then $xyz \equiv o \pmod{60}$.
negative_passages.docid: 92290
negative_passages.text: For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$
negative_passages.docid: 87013
negative_passages.text: If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$