query: there exsit postive integer $x,y$ such $p\mid(x^2+y^2+n)$
positive_passages.docid: 67332
positive_passages.text: Sum of two squares modulo p
negative_passages.docid: 145041
negative_passages.text: Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
negative_passages.docid: 22742
negative_passages.text: Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too
negative_passages.docid: 138846
negative_passages.text: Prove by induction that $3\mid n^4-n^2$ for all $n \in \mathbb{Z^+}$, $n\ge2$.
negative_passages.docid: 62239
negative_passages.text: For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
negative_passages.docid: 127293
negative_passages.text: Prove that $(a-b) \mid (a^n-b^n)$
negative_passages.docid: 83407
negative_passages.text: How to show that that $q \mid p-1$?
negative_passages.docid: 22224
negative_passages.text: Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $.
negative_passages.docid: 24572
negative_passages.text: Does $n \mid 2^{2^n+1}+1$ imply $n \mid 2^{2^{2^n+1}+1}+1$?
negative_passages.docid: 13831
negative_passages.text: Find $ P(Z>X+Y)$ where $X,Y,Z \sim U(0,1)$ independently
negative_passages.docid: 33714
negative_passages.text: $p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
negative_passages.docid: 10236
negative_passages.text: Show that $a^n \mid b^n$ implies $a \mid b$
negative_passages.docid: 31295
negative_passages.text: $x+y\sqrt{2}$ infimum ($x,y\in \mathbb{Z}$)
negative_passages.docid: 153100
negative_passages.text: A counterexample to "$k\mid n$ if and only if $k\mid n^2$"
negative_passages.docid: 83721
negative_passages.text: If a prime $p\mid ab$, then $p\mid a$ or $p\mid b\ $ [Euclid's Lemma]
negative_passages.docid: 136933
negative_passages.text: Which $n$ can be expressed as $x^2 − y^2$?
negative_passages.docid: 50781
negative_passages.text: $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$