query: Let $D$ be an integral domain $a,b \in D$. $m,n$ are relatively prime and $a^n=b^n$ and $a^m=b^m \implies a=b$.
positive_passages.docid: 36602
positive_passages.text: $a^m=b^m$ and $a^n=b^n$ imply $a=b$
negative_passages.docid: 74370
negative_passages.text: $P$ is a prime ideal of $R$ iff $R/P$ is an integral domain. : $P≠R$
negative_passages.docid: 137750
negative_passages.text: Let $(a_n)$ be a strictly increasing sequence of positive integers such that: $a_2 = 2$, $a_{mn} = a_m a_n$ for $m, n$ relatively prime.
negative_passages.docid: 75614
negative_passages.text: Proving that $n|m\implies f_n|f_m$
negative_passages.docid: 98345
negative_passages.text: Characteristic of an integral domain must be either $0$ or a prime number.
negative_passages.docid: 73773
negative_passages.text: $C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.
negative_passages.docid: 135152
negative_passages.text: Non-UFD integral domain such that prime is equivalent to irreducible?
negative_passages.docid: 33645
negative_passages.text: Proving that an integral domain has at most two elements that satisfy the equation $x^2 = 1$.
negative_passages.docid: 119051
negative_passages.text: Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $\text{gcd}(m,n) = 1$, $m$ is prime.
negative_passages.docid: 10236
negative_passages.text: Show that $a^n \mid b^n$ implies $a \mid b$
negative_passages.docid: 148455
negative_passages.text: Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
negative_passages.docid: 19822
negative_passages.text: For $D$ a GCD domain , let $a,b,x \in D \setminus \{0\}$ , then is it true that $\gcd (ax,bx)=x \cdot \gcd (a,b)$?
negative_passages.docid: 19700
negative_passages.text: Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$
negative_passages.docid: 12862
negative_passages.text: Prove that two distinct numbers of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $\gcd=2$ if $a$ is odd
negative_passages.docid: 138058
negative_passages.text: Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
negative_passages.docid: 156966
negative_passages.text: Prove that if $b = dk$ where $gcd(m, a) = d$, there are $d$ solutions to $ax \equiv b \mod m$
negative_passages.docid: 72640
negative_passages.text: $(m,n) = (\gcd(m,n))$ as ideals in $\Bbb Z$