query: If $a,b$ is an $R$-sequence, then $(ax-b)$ is prime
positive_passages.docid: 50665
positive_passages.text: If $a,b$ is an $R$-sequence, then $ax-b$ is prime (Eisenbud, Exercise 10.4)
negative_passages.docid: 344
negative_passages.text: Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
negative_passages.docid: 125499
negative_passages.text: Find this sequence $a_{i}$ such $\gcd{(a_{i},a_{j})}=\gcd{(i,j)}$
negative_passages.docid: 123419
negative_passages.text: Abstract algebra: homomorphism and an element of a prime order
negative_passages.docid: 37330
negative_passages.text: Prove that any two consecutive terms of the Fibonacci sequence are relatively prime
negative_passages.docid: 35731
negative_passages.text: Prove that if $A$ is nonsingular, then the sequence $X_{k+1}=X_k+X_k(I-AX_k)$ converges to $A^{-1}$ if and only if $ρ(I-X_0A)<1$.
negative_passages.docid: 144693
negative_passages.text: Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$
negative_passages.docid: 47856
negative_passages.text: $f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
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: 90144
negative_passages.text: Show that if $a+bi$ is irreducible over $\mathbb{Z} [i]$ then $a^2 + b^2$ is prime over $\mathbb{Z} $
negative_passages.docid: 138669
negative_passages.text: proving that elements of a sequence are pairwise coprime
negative_passages.docid: 116585
negative_passages.text: $r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root
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: 14688
negative_passages.text: Do sequences in $\ell^p$ belong to some $\ell^q$ for $q<p$?
negative_passages.docid: 94515
negative_passages.text: Proving that the elements of a sequence will always be co-prime to each other.
negative_passages.docid: 103147
negative_passages.text: $\operatorname{gcd}(ab,a+b)=1$ if $a$ and $b$ are relatively prime
negative_passages.docid: 149747
negative_passages.text: Prove that if I is a prime ideal of R then I[x] is a prime ideal of R[x]