query: Prove/disprove: $\forall n,a\in\mathbb{Z}: \enspace a^{\varphi(n) + 1} \equiv a \enspace (\text{mod }n)$
positive_passages.docid: 149589
positive_passages.text: Analogous results to Euler's theorem
negative_passages.docid: 21781
negative_passages.text: Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$
negative_passages.docid: 120571
negative_passages.text: What does it mean $a \;\text{mod}\; 1$?
negative_passages.docid: 127816
negative_passages.text: Number theory proof mod p
negative_passages.docid: 62996
negative_passages.text: A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
negative_passages.docid: 154933
negative_passages.text: Prove $(p-k)!(k-1)!\equiv (-1)^k \text{ mod p }$
negative_passages.docid: 118299
negative_passages.text: Show that $(\mathbb{Z}[x]/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$
negative_passages.docid: 17176
negative_passages.text: Proof that $x^n \mod b = (x \mod b)^n$
negative_passages.docid: 49578
negative_passages.text: Proving $a\mathbb{Z}\cap b\mathbb{Z}=[a,b]\mathbb{Z}$
negative_passages.docid: 59134
negative_passages.text: Proving idempotent elements in $\mathbb{Z}_n$
negative_passages.docid: 51465
negative_passages.text: New, elegant proofs for $\varphi(p^{k})=p^{k}-p^{k-1}$
negative_passages.docid: 12210
negative_passages.text: Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
negative_passages.docid: 150007
negative_passages.text: Prove that $(\mathbb{Z}_n , +)$, the integers $\pmod{n}$ under addition, is a group.
negative_passages.docid: 51687
negative_passages.text: Is the set $ \ (\mathbb{Z}_p \setminus \mathbb{Z}) \cap \mathbb{Q}$ non-empty?
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: 5577
negative_passages.text: If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.