query: How many $\alpha \in S_n$ are there with $\alpha^2 = 1$?
positive_passages.docid: 48268
positive_passages.text: How many $\alpha \in S_n$ are such that $\alpha^2 = 1$?
negative_passages.docid: 152172
negative_passages.text: Does there exist a positive irrational number $\alpha $, such that for any positive integer $n$ the number $\lfloor n\alpha \rfloor$ is not a prime?
negative_passages.docid: 81775
negative_passages.text: Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
negative_passages.docid: 59442
negative_passages.text: Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
negative_passages.docid: 24466
negative_passages.text: Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$
negative_passages.docid: 8446
negative_passages.text: Number of solutions over positive integers of $x_1 + x_2 + ... + x_n = S$ for $x_i \in [0, 25]$
negative_passages.docid: 1914
negative_passages.text: General way to determine $\mathbb{Q}(\gamma) = \mathbb{Q}(\alpha,\beta)$ given $\alpha$ and $\beta$
negative_passages.docid: 146083
negative_passages.text: Stirling numbers of the second kind on Multiset
negative_passages.docid: 142038
negative_passages.text: Finding the number of elements of order two in the symmetric group $S_4$
negative_passages.docid: 72410
negative_passages.text: If $\alpha_1,\alpha_2,\ldots,\alpha_n$ be the roots of the equation $x^n+1$
negative_passages.docid: 26343
negative_passages.text: Number of certain (0,1)-matrices, Stanley's Enumerative Combinatorics
negative_passages.docid: 16749
negative_passages.text: Let $A_i= \left \{...,-2,-1,0,1,...,i \right\}$. Find $\bigcup_{i=1}^{n} A_i$ and $\bigcap_{i=1}^{n} A_i$
negative_passages.docid: 109245
negative_passages.text: α² is a cycle if and only if s is odd
negative_passages.docid: 107010
negative_passages.text: How many elements of order $k$ are in $S_n$?
negative_passages.docid: 43219
negative_passages.text: If $[K(\alpha):K]=p\neq q=[K(\beta):K]$ then $[K(\alpha+\beta):K]=pq$
negative_passages.docid: 26109
negative_passages.text: Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$
negative_passages.docid: 75639
negative_passages.text: Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$