query: Find all number $n\in \mathbb N$ such that $\varphi (n)=14$
positive_passages.docid: 113695
positive_passages.text: Show that there is no integer n with $\phi(n)$ = 14
negative_passages.docid: 15268
negative_passages.text: Find all $x,y \in \mathbb N^*$ such that $\frac{1}{x} + \frac{1}{y}=5$
negative_passages.docid: 61858
negative_passages.text: Is there $\phi(n)=n/6$
negative_passages.docid: 97878
negative_passages.text: Are there uncountably many surjections from $\mathbb N$ to $\mathbb N$?
negative_passages.docid: 142008
negative_passages.text: Find all natural numbers $n$ such that ${(n+1)}^{1/2}-{(n-1)}^{1/2}$ is rational
negative_passages.docid: 45460
negative_passages.text: Proof verification of a number theory problem involving sequences.
negative_passages.docid: 17306
negative_passages.text: To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$
negative_passages.docid: 136885
negative_passages.text: Series of natural numbers which has all same digits
negative_passages.docid: 55984
negative_passages.text: How can I find the answer to $\varphi(2^{400} + 2)$?
negative_passages.docid: 20504
negative_passages.text: $\forall m,n \in \Bbb N$ : $\ 56786730\mid mn(m^{60}-n^{60})$
negative_passages.docid: 110958
negative_passages.text: How to solve for any given natural number n?
negative_passages.docid: 149508
negative_passages.text: Prove $9^n \equiv 9 \pmod{24}\quad \forall n \in \Bbb N^*$
negative_passages.docid: 150430
negative_passages.text: Let $S={1, 2, 3, 4, ....., 2070}$ find the number of subsets of $S$ whose sum of elements in $S$ is divisible by 9
negative_passages.docid: 18093
negative_passages.text: Find $x$ such that $\sum_{k=1}^{2014} k^k \equiv x \pmod {10}$
negative_passages.docid: 8884
negative_passages.text: Find all natural numbers $n$ and $m$ for which $n^m=(n-1)!+1$
negative_passages.docid: 96903
negative_passages.text: How to show via inductive proof (and without the use of the binomial theorem) that $ {n\choose k} \in \mathbb{N} $?
negative_passages.docid: 41571
negative_passages.text: Prove that in a sequence of numbers $49 , 4489 , 444889 , 44448889\ldots$