query: Does there exist a positive irrational number $\alpha$, the number $\lfloor n\alpha \rfloor$ is never a prime?
positive_passages.docid: 152172
positive_passages.text: Does there exist a positive irrational number $\alpha $, such that for any positive integer $n$ the number $\lfloor n\alpha \rfloor$ is not a prime?
negative_passages.docid: 35837
negative_passages.text: How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational?
negative_passages.docid: 48268
negative_passages.text: How many $\alpha \in S_n$ are such that $\alpha^2 = 1$?
negative_passages.docid: 53525
negative_passages.text: Infiniteness of non-twin primes.
negative_passages.docid: 5529
negative_passages.text: Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$
negative_passages.docid: 48975
negative_passages.text: Transcendental numbers and irrationality measure
negative_passages.docid: 48587
negative_passages.text: Why is $1$ not a prime number?
negative_passages.docid: 15110
negative_passages.text: About irrational logarithms
negative_passages.docid: 147534
negative_passages.text: The set of irrational numbers is not a $F_{\sigma}$ set.
negative_passages.docid: 51649
negative_passages.text: prove there is no smallest positive rational number
negative_passages.docid: 13592
negative_passages.text: How can $\pi$ be irrational if it can be calculated using an infinite series of rational numbers?
negative_passages.docid: 13550
negative_passages.text: For an irrational number $\alpha$, prove that the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$
negative_passages.docid: 108748
negative_passages.text: Why there are real numbers with infinite digits, but no such natural numbers (or another reason why real numbers are uncountable)
negative_passages.docid: 148035
negative_passages.text: Is it possible for a sequence of rational numbers to have a non rational limit?
negative_passages.docid: 100319
negative_passages.text: Functional equation with strange property about irrational numbers
negative_passages.docid: 44600
negative_passages.text: How can I prove: if $p $ is prime and $n>1$, then $ p^{\frac1n} $ is irrational?
negative_passages.docid: 138446
negative_passages.text: Proving that there exists an irrational number in between any given real numbers