query: Does $\exists n \in \mathbb{Z}$ such that $2^n$ can start with $9786543120$?
positive_passages.docid: 45240
positive_passages.text: Starting digits of $2^n$.
negative_passages.docid: 78726
negative_passages.text: Prove via mathematical induction that $4n < 2^n$ for all $ n≥5$.
negative_passages.docid: 126015
negative_passages.text: Elementary number theory in sets
negative_passages.docid: 116518
negative_passages.text: Proof by induction: $n$th Fibonacci number is at most $ 2^n$
negative_passages.docid: 24403
negative_passages.text: Prove that $ n < 2^{n}$ for all natural numbers $n$.
negative_passages.docid: 135892
negative_passages.text: Understanding the proof for showing that there are infinitely many $n\in\mathbb{Z}^+$ such that $n!+1$ is divisible by at least two distinct primes
negative_passages.docid: 146980
negative_passages.text: Zero divisors of $\mathbb{Z}×\mathbb{Z}×\mathbb{Z}$
negative_passages.docid: 30374
negative_passages.text: If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.
negative_passages.docid: 87267
negative_passages.text: Conjecture similar to Fermat's Theorem.
negative_passages.docid: 125604
negative_passages.text: Proof that $\mathbb{Z}$ has no zero divisors
negative_passages.docid: 83909
negative_passages.text: Is $2^{2n} = O(2^n)$?
negative_passages.docid: 75373
negative_passages.text: Show that $2^n>n$ for $n\in \mathbb{N}$
negative_passages.docid: 141799
negative_passages.text: Is $2^{|\mathbb{N}|} = |\mathbb{R}|$?
negative_passages.docid: 16507
negative_passages.text: Is $\mathbb{Z}= \{\dots -3, -2, -1, 0 ,1 ,2 , 3, \dots \}$ countable?
negative_passages.docid: 104841
negative_passages.text: Can $\sin n$ get arbitrarily close to $1$ for $n\in\mathbb{N}?$
negative_passages.docid: 150677
negative_passages.text: What are the integers $n$ such that $\mathbb{Z}[\sqrt{n}]$ is integrally closed?
negative_passages.docid: 21781
negative_passages.text: Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$