query: Motivation for proof of $(((\phi \to \psi) \to \phi) \to \phi)$
positive_passages.docid: 31299
positive_passages.text: Using the Fitch system, how do I prove $((p \implies q) \implies p)\implies p$?
negative_passages.docid: 48291
negative_passages.text: Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
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: 73095
negative_passages.text: Foundational proof for Mersenne primes
negative_passages.docid: 135302
negative_passages.text: Proof on Euler totient function
negative_passages.docid: 17178
negative_passages.text: Order of $\phi(g)$ divides the order of $g$
negative_passages.docid: 13136
negative_passages.text: Element chasing proof that $(A\setminus C)\setminus(B\setminus C) = (A\setminus B)\setminus C$
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: 54244
negative_passages.text: Understanding the Proof of L'Hopital's Rule
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: 4512
negative_passages.text: Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
negative_passages.docid: 22842
negative_passages.text: The Interpretation of $|\psi|^2$ and the No-Clone Theorem
negative_passages.docid: 49938
negative_passages.text: Proof of $a^{\phi (n) +1} \equiv a \pmod{\! n} $ Carmichael generalization of Fermat-Euler's theorem.
negative_passages.docid: 125188
negative_passages.text: how to prove $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod{mn}$ where m and n are relatively prime?
negative_passages.docid: 45179
negative_passages.text: When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
negative_passages.docid: 137711
negative_passages.text: Alternative proof that $[\Bbb{Q}(\zeta_n) : \Bbb{Q} ]= \varphi(n)$ uses circular reasoning?
negative_passages.docid: 76347
negative_passages.text: Shorter proof for some equvalences