query: Prove that $m^{\varphi(N)+1} \equiv m \pmod N$ when $N=pq$, p,q are primes.
positive_passages.docid: 49938
positive_passages.text: Proof of $a^{\phi (n) +1} \equiv a \pmod{\! n} $ Carmichael generalization of Fermat-Euler's theorem.
negative_passages.docid: 5841
negative_passages.text: Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
negative_passages.docid: 5708
negative_passages.text: Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
negative_passages.docid: 157711
negative_passages.text: Number Theory: Prove that $m | \phi(a^m - 1)$
negative_passages.docid: 44729
negative_passages.text: Prove $g$ is a generator if $g^q=1 \pmod p$
negative_passages.docid: 27170
negative_passages.text: $a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
negative_passages.docid: 3109
negative_passages.text: Prove that $(p+q)^m \leq p^m+q^m$
negative_passages.docid: 103732
negative_passages.text: Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
negative_passages.docid: 39843
negative_passages.text: Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$
negative_passages.docid: 27784
negative_passages.text: Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
negative_passages.docid: 123033
negative_passages.text: Showing a number $n$ is prime if $(n-2)! \equiv 1 \pmod n$
negative_passages.docid: 135986
negative_passages.text: Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
negative_passages.docid: 59566
negative_passages.text: Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
negative_passages.docid: 88656
negative_passages.text: For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$
negative_passages.docid: 125868
negative_passages.text: Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
negative_passages.docid: 127816
negative_passages.text: Number theory proof mod p