query: Prove that $a^2$ when divided by $5$ cannot have a remainder of $3$.
positive_passages.docid: 144802
positive_passages.text: Prove no solution to $x^2\equiv3\bmod5$
negative_passages.docid: 145558
negative_passages.text: Prove that $2^n+3^n $ is never a perfect square
negative_passages.docid: 115460
negative_passages.text: Proving $a\mid b^2,\,b^2\mid a^3,\,a^3\mid b^4,\ldots\implies a=b$ - why is my approach incorrect
negative_passages.docid: 107725
negative_passages.text: Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
negative_passages.docid: 21906
negative_passages.text: Show $3^m + 3^n +1$ cannot be a perfect square for $m,n$ being positive integers.
negative_passages.docid: 83662
negative_passages.text: Prove that $ a $ divides $1^{a^{n}} + 2^{a^{n}} + ... + (a-1)^{a^{n}}$?
negative_passages.docid: 109532
negative_passages.text: Proving $5^n \equiv 1 \pmod {2^r}$ when $n=2^{r-2}$
negative_passages.docid: 139755
negative_passages.text: Prove that $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational. Generalise this.
negative_passages.docid: 44307
negative_passages.text: Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$
negative_passages.docid: 7491
negative_passages.text: How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$?
negative_passages.docid: 78416
negative_passages.text: Show that $7$ divides $a^{15}-a^3$ for any integer $a$
negative_passages.docid: 84379
negative_passages.text: Prove that there is no integer a for which $a^2 - 3a - 19$ is divisible by 289
negative_passages.docid: 149071
negative_passages.text: Let $d$ be any positive integer not equal to $2, 5,$ or $ 13$ , then $\exists a, b \in \{2, 5, 13, d\}$ such that $ab − 1$ is not a perfect square?
negative_passages.docid: 76032
negative_passages.text: Prove that if m is a square integer then m is neither congruent to 2 modulo 5 nor congruent to 3 modulo 5
negative_passages.docid: 137808
negative_passages.text: $n^2 + 3n +5$ is not divisible by $121$
negative_passages.docid: 103803
negative_passages.text: Prove from first princibles $\frac{x^2-4}{x-4} \rightarrow -5$ as $x \rightarrow 3$
negative_passages.docid: 22973
negative_passages.text: Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares.