query: Show that for any open subset of $\Bbb R$, there is a fraction with prime terms that belongs to it
positive_passages.docid: 48264
positive_passages.text: Are fractions with prime numerator and denominator dense?
negative_passages.docid: 45405
negative_passages.text: An elementary way to show any bounded subset of $\Bbb{R}^k$ is totally bounded
negative_passages.docid: 70066
negative_passages.text: Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
negative_passages.docid: 5978
negative_passages.text: Prove that there exists only one prime number of the form $p^2−1$ where $p≥2$ is an integer
negative_passages.docid: 5360
negative_passages.text: Ideal contained in a finite union of prime ideals
negative_passages.docid: 83721
negative_passages.text: If a prime $p\mid ab$, then $p\mid a$ or $p\mid b\ $ [Euclid's Lemma]
negative_passages.docid: 11817
negative_passages.text: Proving that every set $A \subset \Bbb N$ of size $n$ contains a subset $B \subset A$ with $n | \sum_{b \in B} b$
negative_passages.docid: 65764
negative_passages.text: Partitioning $\{1,2,\ldots,k\}$ into $p$ subsets with equal sums, where $p$ is prime
negative_passages.docid: 108380
negative_passages.text: Assume $P$ is a prime ideal s.t. $K \subset P$, show $f(P)$ is a prime ideal
negative_passages.docid: 136106
negative_passages.text: Is every open subset of $ \mathbb{R} $ uncountable?
negative_passages.docid: 37330
negative_passages.text: Prove that any two consecutive terms of the Fibonacci sequence are relatively prime
negative_passages.docid: 72640
negative_passages.text: $(m,n) = (\gcd(m,n))$ as ideals in $\Bbb Z$
negative_passages.docid: 6100
negative_passages.text: Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
negative_passages.docid: 69867
negative_passages.text: For each $\beta \in(0,1)$, give a sequence $\{a_j\}\subset(0,1)$ with $\prod (1-\alpha_j) = \beta $
negative_passages.docid: 73095
negative_passages.text: Foundational proof for Mersenne primes
negative_passages.docid: 121890
negative_passages.text: Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
negative_passages.docid: 110022
negative_passages.text: Showing prime and maximal ideal