query: Is $\phi(n) + \sigma(n) \geq 2n$ always true?
positive_passages.docid: 148901
positive_passages.text: Can a composite number $n$ be the arithmetic mean of $\sigma(n)$ and $\varphi(n)$?
negative_passages.docid: 54791
negative_passages.text: Is it true or false that $\lvert P([0, 1]) \setminus P((0, 1)) \rvert > \lvert P(\mathbb{N}) \rvert$
negative_passages.docid: 102406
negative_passages.text: Why is it not true that if $\Phi \models \exists x \phi$ then there is a term $t$ with $\Phi \models \phi \frac{t}{x}$?
negative_passages.docid: 22129
negative_passages.text: Why is $\sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|$ true?
negative_passages.docid: 98662
negative_passages.text: $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$
negative_passages.docid: 25419
negative_passages.text: If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$
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: 44575
negative_passages.text: The sigma function (sum of divisors) multiplicative proof
negative_passages.docid: 141167
negative_passages.text: What is $\int_{c}^{\infty}\Phi(a+bx) \phi( x ) dx$ for $c\in\mathbb{R}$
negative_passages.docid: 34491
negative_passages.text: Is there a way to show that $\sqrt{p_{n}} < n$?
negative_passages.docid: 79011
negative_passages.text: If $(F:E)<\infty$, is it always true that $\operatorname{Aut}(F/E)\leq(F:E)?$
negative_passages.docid: 28332
negative_passages.text: Proof of $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$
negative_passages.docid: 137982
negative_passages.text: Show that $\textbf{ZF}$ shows $\phi$ is $\Delta_1$
negative_passages.docid: 37346
negative_passages.text: If $m, n$ be positive integers, prove that $\phi(mn)=\phi((m,n))\phi([m,n])$, where $(m,n)=$ gcd of $m, n$ and $[m, n]=$ lcm of $m, n$.
negative_passages.docid: 9753
negative_passages.text: How to solve the equation $\phi(n) = k$?
negative_passages.docid: 45179
negative_passages.text: When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
negative_passages.docid: 119118
negative_passages.text: Prove $n\mid \phi(2^n-1)$