query: Prove: For $m,n,a \in \mathbb{Z+}$, if $m \gt n$, then $a^{2^n} +1 \mid a^{2^m} -1$
positive_passages.docid: 22093
positive_passages.text: How to show $a^{2^n}+1 \mid a^{2^m}-1$?
negative_passages.docid: 92975
negative_passages.text: Does $a^n \mid b^n$ imply $a\mid b$?
negative_passages.docid: 12100
negative_passages.text: $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $n \mid \frac{a^{n}-b^{n}}{a-b}$
negative_passages.docid: 10236
negative_passages.text: Show that $a^n \mid b^n$ implies $a \mid b$
negative_passages.docid: 76696
negative_passages.text: Using induction to prove $2^{n-1}(1 + a_1a_2\ldots a_n) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ for $a_i \geq 1$
negative_passages.docid: 55428
negative_passages.text: Inductive proof of $a_n = 3*2^{n-1} + 2(-1)^n$ if $a_n = a_{n-1} + 2*a_{n-2}, a_1 = 1, a_2 = 8$
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 6012
negative_passages.text: Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
negative_passages.docid: 12096
negative_passages.text: If $a+1/a$ is an integer, then so is $a^t+1/a^t$ for $t\in\mathbb N$
negative_passages.docid: 87522
negative_passages.text: If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$
negative_passages.docid: 35945
negative_passages.text: Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
negative_passages.docid: 9846
negative_passages.text: Prove $n-m \mid n^r - m^r\,$ [Factor Theorem, monomial case]
negative_passages.docid: 138894
negative_passages.text: Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
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: 130488
negative_passages.text: Prove by induction that $3\mid n^3 - n$
negative_passages.docid: 114780
negative_passages.text: $\forall a,b,c \in \mathbb{N}\!:\ a\mid b \Rightarrow ac\mid bc. $ Divisibility Proof
negative_passages.docid: 153313
negative_passages.text: Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.