query: show that $ n | \phi (a ^ n-1) $
positive_passages.docid: 157711
positive_passages.text: Number Theory: Prove that $m | \phi(a^m - 1)$
negative_passages.docid: 53263
negative_passages.text: Show $\sum\limits_{d|n}\phi(d) = n$.
negative_passages.docid: 138819
negative_passages.text: Prove that $\phi(n) \geq \sqrt{n}/2$
negative_passages.docid: 135808
negative_passages.text: Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
negative_passages.docid: 113695
negative_passages.text: Show that there is no integer n with $\phi(n)$ = 14
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: 98017
negative_passages.text: If $a_{n+1}/a_n\to\ell$ show $a_n^{1/n}\to\ell$
negative_passages.docid: 81346
negative_passages.text: Let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove $r \mid \phi(n)$.
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: 91197
negative_passages.text: Proving $n! \ge 2^{n-1 }$for all $n\ge1 $by mathematical Induction
negative_passages.docid: 12301
negative_passages.text: Show that $\lim_{n \rightarrow \infty} \sqrt[n]{a} =1$
negative_passages.docid: 88662
negative_passages.text: $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
negative_passages.docid: 46741
negative_passages.text: Prove that $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer without using concepts of abstract algebra
negative_passages.docid: 84021
negative_passages.text: How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?
negative_passages.docid: 33344
negative_passages.text: A different way to show that $a^k-1 \mid a^n-1$ if $k\mid n$?
negative_passages.docid: 94558
negative_passages.text: Show that $\phi(mn) = \phi(m)\phi(n)\frac{d}{\phi(d)}$
negative_passages.docid: 123275
negative_passages.text: Prove $[A,B^n] = nB^{n-1}[A,B]$