query: The periodic orbits in $3x+1$ conjecture on $\mathbb {Z}$
positive_passages.docid: 27448
positive_passages.text: The Collatz conjecture algorithm applied to negative integers
negative_passages.docid: 25290
negative_passages.text: Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$
negative_passages.docid: 78118
negative_passages.text: Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$?
negative_passages.docid: 127365
negative_passages.text: Question on showing a bijection between $\pi_1(X,x_0)$ and $[S^1, X]$ when X is path connected.
negative_passages.docid: 100696
negative_passages.text: Let $p$ be prime. Consider the sequence: $1, 2^2, 3^3, 4^4, 5^5, 6^6 \cdots \pmod p$. Prove the sequence is periodic with smallest period $p(p-1).$
negative_passages.docid: 79144
negative_passages.text: Prove that the orbit of an iterated rotation of 0 (by (A)(Pi), A irrational) around a circle centered at the origin is dense in the circle.
negative_passages.docid: 28934
negative_passages.text: Presentation of SL$(n,\mathbb{Z}_p)$
negative_passages.docid: 68915
negative_passages.text: What are the units in $\mathbb{Z}[\root 3 \of 2]$?
negative_passages.docid: 110167
negative_passages.text: A conjecture with connection to the $abc$-conjecture
negative_passages.docid: 33781
negative_passages.text: Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$?
negative_passages.docid: 69694
negative_passages.text: Classify orbits of conjugating action on $GL_2(\mathbb{C})$
negative_passages.docid: 84823
negative_passages.text: When is the group of units in $\mathbb{Z}_n$ cyclic?
negative_passages.docid: 35691
negative_passages.text: $\mathbb Z[\sqrt 3]$ contains infinitely many units
negative_passages.docid: 93169
negative_passages.text: Show that $(\mathbb Z[x],+)$ and $(\mathbb Q_{>0},\cdot)$ are isomorphic groups
negative_passages.docid: 55673
negative_passages.text: Alternate proof that $\exists\ x \in \Bbb{Z}$ such that $ \gcd (a+bx,c) = 1$?
negative_passages.docid: 158235
negative_passages.text: Laurent Series for $\cot(\pi z)$
negative_passages.docid: 99782
negative_passages.text: $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$