query: Show that if $n^2$ is even then $n$ is even, for $n$ an integer.
positive_passages.docid: 153532
positive_passages.text: Proving $a^2$ is even $\implies$ $a$ is even by contradiction
negative_passages.docid: 123267
negative_passages.text: Proving that even numbers equal the sum of two odd numbers.
negative_passages.docid: 2554
negative_passages.text: Show that if $n>2$, then $(n!)^2>n^n$.
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: 135351
negative_passages.text: $C(n,p)$: even or odd?
negative_passages.docid: 44182
negative_passages.text: Proving that an integer is even if and only if it is not odd
negative_passages.docid: 75373
negative_passages.text: Show that $2^n>n$ for $n\in \mathbb{N}$
negative_passages.docid: 85731
negative_passages.text: Prove by induction that $n! > n^2$
negative_passages.docid: 68480
negative_passages.text: Proof by Induction: Prove that $2^n > n^2$, for all natural numbers greater than or equal to $5$
negative_passages.docid: 4970
negative_passages.text: For $n \geq 2$, show that $n \nmid 2^{n}-1$
negative_passages.docid: 15633
negative_passages.text: If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
negative_passages.docid: 18953
negative_passages.text: Prove that even + odd is odd.
negative_passages.docid: 54369
negative_passages.text: Proof of $n^2 \leq 2^n$.
negative_passages.docid: 156641
negative_passages.text: Show that $\gcd(3n,3n+ 2) = 1$ when $n$ is odd
negative_passages.docid: 103538
negative_passages.text: Partitions of $n$ into distinct odd and even parts proof
negative_passages.docid: 135018
negative_passages.text: Proving that a sequence has $a_n = a_{n+2}$ for $n$ sufficiently large.
negative_passages.docid: 50534
negative_passages.text: Prove that if $n^2$ is divided by 3, then also $n$ can also be divided by 3.