query: $(n-1)^2\vert n^k-1$ if and only if $n-1\vert k$
positive_passages.docid: 143543
positive_passages.text: Prove that $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$
negative_passages.docid: 142008
negative_passages.text: Find all natural numbers $n$ such that ${(n+1)}^{1/2}-{(n-1)}^{1/2}$ is rational
negative_passages.docid: 91197
negative_passages.text: Proving $n! \ge 2^{n-1 }$for all $n\ge1 $by mathematical Induction
negative_passages.docid: 108318
negative_passages.text: counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
negative_passages.docid: 95862
negative_passages.text: Prove that $t_{n-1, \alpha/2}$ is strictly decreasing in $n$
negative_passages.docid: 77076
negative_passages.text: Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
negative_passages.docid: 100558
negative_passages.text: Prove that for any even positive integer $n$, $n^2-1 \mid 2^{n!}-1$
negative_passages.docid: 126574
negative_passages.text: Proving $\prod_k \sin \pi k / n = n / 2^{n-1}$
negative_passages.docid: 115459
negative_passages.text: Proving this $F_{n+1} \cdot F_{n-1} - F^2_n = (-1)^n$ by induction
negative_passages.docid: 136077
negative_passages.text: Prove that $x-1$ is a factor of $x^n-1$
negative_passages.docid: 27063
negative_passages.text: Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
negative_passages.docid: 151520
negative_passages.text: About the parity of the product $(a_1-1)(a_2-2)\cdots(a_n-n)$
negative_passages.docid: 142534
negative_passages.text: Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
negative_passages.docid: 4970
negative_passages.text: For $n \geq 2$, show that $n \nmid 2^{n}-1$
negative_passages.docid: 35553
negative_passages.text: If $n$ is composite, so is $2^n-1$
negative_passages.docid: 148330
negative_passages.text: Induction Proof for $F_{2n} = F^2_{n+1} - F^2_{n-1}$
negative_passages.docid: 22073
negative_passages.text: Prove by induction that $n^2<n!$