query: Show that for each $m\in \mathbb{N}$, there exists $n\in \mathbb{N}$ such that $x^2\equiv 1 \pmod n$ has at least $m$ solutions?
positive_passages.docid: 112390
positive_passages.text: Solutions of $x^2 \equiv 1 \pmod N$
negative_passages.docid: 152591
negative_passages.text: For which prime numbers $p$ does the congruence $x^2 + x + 1 \equiv 0 \pmod{p}$ have solutions?
negative_passages.docid: 17601
negative_passages.text: How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
negative_passages.docid: 144851
negative_passages.text: For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
negative_passages.docid: 24820
negative_passages.text: Prove that if $n$ is a perfect square, then either $n \equiv 1 \pmod 4$, or $n \equiv 0 \pmod 4$.
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
negative_passages.docid: 94965
negative_passages.text: Solve $x\equiv 1\pmod2$, $x\equiv 2\pmod3$, $x\equiv 3\pmod4$, $x\equiv 4\pmod5$, $x\equiv 5\pmod6$ and $x\equiv 0\pmod7$
negative_passages.docid: 107990
negative_passages.text: Does there exist an integer $x$ satisfying the following congruences?
negative_passages.docid: 80843
negative_passages.text: Find all solutions to $x^9 \equiv 25 \pmod {29}$
negative_passages.docid: 26478
negative_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: 46147
negative_passages.text: Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$
negative_passages.docid: 978
negative_passages.text: Solving $ax \equiv c \pmod b$ efficiently when $a,b$ are not coprime
negative_passages.docid: 148455
negative_passages.text: Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
negative_passages.docid: 29204
negative_passages.text: $a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus
negative_passages.docid: 7466
negative_passages.text: How would I prove that if an integer $n>2$, then $\bar{z}=z^{n-1}$ has $n+1$ solutions
negative_passages.docid: 21250
negative_passages.text: Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
negative_passages.docid: 96185
negative_passages.text: Solutions of $x^2+y^2+z^2 = n$ in $\mathbb{Z}_2$.