query: If $m$ and $n$ are relatively prime positive integers, prove that $m^{\phi(n)}+n^{\phi(m)}\equiv1\pmod{mn}$
positive_passages.docid: 97196
positive_passages.text: $k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}\,$ if $\,\gcd(l,k)=1$
negative_passages.docid: 63769
negative_passages.text: Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$)
negative_passages.docid: 46065
negative_passages.text: How do you find all $n$ such that $\phi(n)|n$
negative_passages.docid: 124685
negative_passages.text: How to show $(mn)!$ is divisible by $(m!)^n$?
negative_passages.docid: 4512
negative_passages.text: Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
negative_passages.docid: 120629
negative_passages.text: Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them
negative_passages.docid: 158582
negative_passages.text: Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$
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: 62382
negative_passages.text: I suppose $(a,m)=(a-1,m)=1$, show that $1+a+a^2+\ldots+a^{\phi(m)-1}\equiv0\pmod m$
negative_passages.docid: 144819
negative_passages.text: Proving $P(n) =n^{\phi(n)} \prod\limits_{d \mid n} \left(\frac{d!}{d^d} \right)^{\mu(n/d)}$
negative_passages.docid: 6700
negative_passages.text: An Olympiad problem in prime numbers :can we proof that "n" is prime in this case "
negative_passages.docid: 68385
negative_passages.text: Prove that $ \frac{n}{\phi(n)} = \sum\limits_{d \mid n} \frac{\mu^2(d)}{\phi(d)} $
negative_passages.docid: 123891
negative_passages.text: There are finitely many maps on nonnegative integers satisfying $\phi(ab)=\phi(a)+\phi(b)$
negative_passages.docid: 106401
negative_passages.text: Prove: $p-mq | f(m)$ where $m$ is any integer
negative_passages.docid: 60305
negative_passages.text: Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
negative_passages.docid: 33144
negative_passages.text: Criterion for $\phi(mn)=m\phi(n)$ to be true
negative_passages.docid: 119051
negative_passages.text: Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $\text{gcd}(m,n) = 1$, $m$ is prime.