query: Show that $2 ^ n - 1$ divides $3 ^ n -1$ if and only if $n = 1$
positive_passages.docid: 151692
positive_passages.text: When does $2^n-1$ divide $3^n-1$?
negative_passages.docid: 128306
negative_passages.text: Show that for all $n ∈ Z$, $2n^3 + 3n^2 + n$ is divisible by both 2 and 3.
negative_passages.docid: 78416
negative_passages.text: Show that $7$ divides $a^{15}-a^3$ for any integer $a$
negative_passages.docid: 1448
negative_passages.text: $x^a - 1$ divides $x^b - 1$ if and only if $a$ divides $b$
negative_passages.docid: 82774
negative_passages.text: Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$
negative_passages.docid: 136077
negative_passages.text: Prove that $x-1$ is a factor of $x^n-1$
negative_passages.docid: 33344
negative_passages.text: A different way to show that $a^k-1 \mid a^n-1$ if $k\mid n$?
negative_passages.docid: 133017
negative_passages.text: Prove that $n^n>1\times3\times5\times7\times \dots\times(2n-1)$
negative_passages.docid: 129238
negative_passages.text: Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.
negative_passages.docid: 141088
negative_passages.text: Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.
negative_passages.docid: 100305
negative_passages.text: Prove $2^n-1$ is divisible by $(2^m-1)^2$ iff...
negative_passages.docid: 50670
negative_passages.text: Use Mathematical Induction to prove 6 divides $n(n+1)(2n+1)$
negative_passages.docid: 55437
negative_passages.text: When does $2^{n}$ divide $3^{n}-1$
negative_passages.docid: 68561
negative_passages.text: How to show $\lceil ( \sqrt3 +1)^{2n}\rceil$ where $n \in \mathbb{ N}$ is divisible by $2^{n+1}$
negative_passages.docid: 94516
negative_passages.text: If $3$ divides $a^2 + b^2$, then $3$ divides $a$ and $3$ divides $b$
negative_passages.docid: 35553
negative_passages.text: If $n$ is composite, so is $2^n-1$
negative_passages.docid: 19202
negative_passages.text: $\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ if and only if $d$ divides $n$