query: Prove that the square of an integer $a$ is of the form $a^2=3k$, or $a^2=3k+1$, where $k\in \mathbb{Z} $
positive_passages.docid: 139
positive_passages.text: Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)
negative_passages.docid: 8087
negative_passages.text: Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer
negative_passages.docid: 49578
negative_passages.text: Proving $a\mathbb{Z}\cap b\mathbb{Z}=[a,b]\mathbb{Z}$
negative_passages.docid: 72470
negative_passages.text: Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
negative_passages.docid: 67605
negative_passages.text: Different proof that $\sqrt{2}$ is irrational
negative_passages.docid: 60348
negative_passages.text: How do I prove $F(a)=F(a^2)?$
negative_passages.docid: 138894
negative_passages.text: Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
negative_passages.docid: 98440
negative_passages.text: Show that any non-negative rational integer may be written as the sum of two squares if it is of the form....
negative_passages.docid: 103360
negative_passages.text: Prove $4p-3$ is a square knowing that $n\mid p-1$ and $p\mid n^3-1$, $p$ prime
negative_passages.docid: 125702
negative_passages.text: There is a square $Q$ consisting of $(0,0), (2,0), (0,2), (2,2)$
negative_passages.docid: 82777
negative_passages.text: If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$?
negative_passages.docid: 86378
negative_passages.text: Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$
negative_passages.docid: 62747
negative_passages.text: Use induction to prove a product of sums of squares is a sum of squares
negative_passages.docid: 91113
negative_passages.text: $ \exists a, b \in \mathbb{Z} $ such that $ a^2 + b^2 = 5^k $
negative_passages.docid: 6169
negative_passages.text: If $ A^3=A$ prove that $Ker\left(A-I\right)+Im\left(A-I\right)=V$
negative_passages.docid: 120954
negative_passages.text: Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
negative_passages.docid: 130027
negative_passages.text: Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$