query: The square of a number is even if and only if the number itself is even
positive_passages.docid: 17047
positive_passages.text: Prove that if n is even then $n^2$ is even and if n is odd then $n^2$ is odd?
negative_passages.docid: 135458
negative_passages.text: When is a sum of consecutive squares equal to a square?
negative_passages.docid: 78407
negative_passages.text: Are there infinitely-many numbers that are both square and triangular?
negative_passages.docid: 70619
negative_passages.text: Supposed $a,b \in \mathbb{Z}$. If $ab$ is odd, then $a^{2} + b^{2}$ is even.
negative_passages.docid: 18953
negative_passages.text: Prove that even + odd is odd.
negative_passages.docid: 156577
negative_passages.text: The cube of any number not a multiple of $7$, will equal one more or one less than a multiple of $7$
negative_passages.docid: 79103
negative_passages.text: Check if a number is a perfect square
negative_passages.docid: 156242
negative_passages.text: As many even numbers as natural numbers.
negative_passages.docid: 79969
negative_passages.text: Proof that a formula $\theta$ is valid if and only if $\forall x \theta$ is valid
negative_passages.docid: 153100
negative_passages.text: A counterexample to "$k\mid n$ if and only if $k\mid n^2$"
negative_passages.docid: 124374
negative_passages.text: A bijective function between a square and its side
negative_passages.docid: 57250
negative_passages.text: Proving that the sum and difference of two squares (not equal to zero) can't both be squares.
negative_passages.docid: 19588
negative_passages.text: $n!+1$ being a perfect square
negative_passages.docid: 158308
negative_passages.text: Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$
negative_passages.docid: 152882
negative_passages.text: How to tell if a Fibonacci number has an even or odd index
negative_passages.docid: 114542
negative_passages.text: Can a number with $100$ $0$'s $100$ $1$'s and $100$ $2$'s ever be a perfect square?
negative_passages.docid: 138081
negative_passages.text: How to determine whether a number can be written as a sum of two squares?