query: $|\{1≤x≤p^2:p^2│x^{p-1}-1\}|=p-1$
positive_passages.docid: 26478
positive_passages.text: Show that the congruence $x^{p-1} \equiv 1 \pmod {p^2}$ has precisely $p-1$ solutions modulo $p^2$
negative_passages.docid: 46760
negative_passages.text: To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$
negative_passages.docid: 155190
negative_passages.text: Stone-Weierstrass theorem with $p(1/x)$
negative_passages.docid: 64096
negative_passages.text: Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.
negative_passages.docid: 59442
negative_passages.text: Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
negative_passages.docid: 137511
negative_passages.text: Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$
negative_passages.docid: 139930
negative_passages.text: Find all polynomials $p$ such that $p(x^2)=p(x)p(x+1)$
negative_passages.docid: 50781
negative_passages.text: $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
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: 3273
negative_passages.text: Trouble in proving that $\|x\|_p = \max|x_j|$
negative_passages.docid: 142785
negative_passages.text: How can I prove Holder inequality for $0<p<1$?
negative_passages.docid: 44019
negative_passages.text: Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
negative_passages.docid: 108995
negative_passages.text: Prove $|e^{i\theta} -1| \leq |\theta|$
negative_passages.docid: 154933
negative_passages.text: Prove $(p-k)!(k-1)!\equiv (-1)^k \text{ mod p }$
negative_passages.docid: 34323
negative_passages.text: Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
negative_passages.docid: 114346
negative_passages.text: Why is $a^{-1}$ mod $p$ equal to $a^{p-2}$ mod $p$?
negative_passages.docid: 26658
negative_passages.text: A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.