query: Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$
positive_passages.docid: 9997
positive_passages.text: For what powers $k$ is the polynomial $n^k-1$ divisible by $(n-1)^2$?
negative_passages.docid: 73102
negative_passages.text: Given the factors of $N$, is there a method for computing the factors of $N-1$ or $N+1$?
negative_passages.docid: 58633
negative_passages.text: Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$
negative_passages.docid: 38364
negative_passages.text: Showing that $x^n -2$ is irreducible in $\mathbb{Q}[X]$
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 143543
negative_passages.text: Prove that $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$
negative_passages.docid: 89381
negative_passages.text: Prove that a real polynomial $x^n+ a_1x^{n-1}+ \cdots a_n$ cannot be completely resolved into linear factors if $a_1^2<a_2$.
negative_passages.docid: 155995
negative_passages.text: Fibonacci proof question: $f_{n+1}f_{n-1}-f_n^2=(-1)^n$
negative_passages.docid: 57416
negative_passages.text: Fibonacci identity: $f_{n-1}f_{n+1} - f_{n}^2 = (-1)^n$
negative_passages.docid: 5043
negative_passages.text: Show that $2^n>1+n\sqrt{2^{n-1}}$
negative_passages.docid: 99052
negative_passages.text: Prove that $( 1 + n^{-2}) ^n \to 1$.
negative_passages.docid: 39739
negative_passages.text: Fibonacci identity: $f_{n+1}f_{n-1} = f_n^2 + (-1)^n$
negative_passages.docid: 103854
negative_passages.text: Showing that $Q_n=D_n+D_{n-1}$
negative_passages.docid: 26011
negative_passages.text: To show that $2$ | $x^{2n} - x^n$, $n \in \mathbb{Z^+}$ is the following adequate?
negative_passages.docid: 51952
negative_passages.text: Show that $x^{n-1}+\cdots +x+1$ is irreducible over $\mathbb Z$ if and only if $n$ is a prime.
negative_passages.docid: 96152
negative_passages.text: Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
negative_passages.docid: 111505
negative_passages.text: Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$