query: For sufficiently large $n$, Which number is bigger, $2^n$ or $n^{1000}$?
positive_passages.docid: 6310
positive_passages.text: How to prove that exponential grows faster than polynomial?
negative_passages.docid: 29578
negative_passages.text: Why is $\binom{2n}{n} \asymp \Theta \big(\frac{2^{2n}}{\sqrt{n}}\big)$?
negative_passages.docid: 26062
negative_passages.text: Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger?
negative_passages.docid: 52542
negative_passages.text: For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?
negative_passages.docid: 54672
negative_passages.text: If 32-bit machines can only handle numbers up to 2^32, why can I write 1000000000000 (trillion) without my machine crashing?
negative_passages.docid: 8815
negative_passages.text: What is the difference between Θ(n) and O(n)?
negative_passages.docid: 130093
negative_passages.text: Prove by Induction : $n < 2^n$
negative_passages.docid: 55437
negative_passages.text: When does $2^{n}$ divide $3^{n}-1$
negative_passages.docid: 4970
negative_passages.text: For $n \geq 2$, show that $n \nmid 2^{n}-1$
negative_passages.docid: 148803
negative_passages.text: 501 distinct coprime integers between 1 and 1000?
negative_passages.docid: 131385
negative_passages.text: How to provide a counterexample for if $n$ is prime, then $2^n -1$ is prime
negative_passages.docid: 19254
negative_passages.text: How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers?
negative_passages.docid: 45240
negative_passages.text: Starting digits of $2^n$.
negative_passages.docid: 77877
negative_passages.text: Is there a domain "larger" than (i.e., a supserset of) the complex number domain?
negative_passages.docid: 56574
negative_passages.text: For any integer $a,b$ let $N_{a,b}$ denote the number of positive integer $x<1000$ satisfying $x= a( mod\;27)$ and $x=b(mod\;37)$. Then
negative_passages.docid: 37525
negative_passages.text: Maximum power of $2$ which divides $3^{1024}-1$
negative_passages.docid: 86168
negative_passages.text: Is $2^k = 2013...$ for some $k$?