query: Let $n$ be a positive integer. Show that the smallest integer greater than $(\sqrt{3} + 1)^{2n}$ is divisible by $2^{n+1}.$
positive_passages.docid: 68561
positive_passages.text: How to show $\lceil ( \sqrt3 +1)^{2n}\rceil$ where $n \in \mathbb{ N}$ is divisible by $2^{n+1}$
negative_passages.docid: 126367
negative_passages.text: Let $p$, $q$ be prime numbers such that $n^{3pq} – n$ is a multiple of $3pq$ for all positive integers $n$. Find the least possible value of $p + q$.
negative_passages.docid: 127833
negative_passages.text: Proving that $7^n(3n+1)-1$ is divisible by 9
negative_passages.docid: 49096
negative_passages.text: How can I prove $5^{3^n}+1$ is divisible by $3^{n+1}$
negative_passages.docid: 26336
negative_passages.text: Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
negative_passages.docid: 5043
negative_passages.text: Show that $2^n>1+n\sqrt{2^{n-1}}$
negative_passages.docid: 76977
negative_passages.text: Prove that $f(n)$ is the nearest integer to $\frac12(1+\sqrt2)^{n+1}$?
negative_passages.docid: 4315
negative_passages.text: show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
negative_passages.docid: 77816
negative_passages.text: Proof problem: Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer.
negative_passages.docid: 51649
negative_passages.text: prove there is no smallest positive rational number
negative_passages.docid: 96332
negative_passages.text: Prove by induction that $\sqrt[n+1]{n+1}<\sqrt[n]{n}$ for $n\ge 3$
negative_passages.docid: 1507
negative_passages.text: The product of $n$ consecutive integers is divisible by $n$ factorial
negative_passages.docid: 44857
negative_passages.text: Why the $(n+1)$th prime number cannot exceed the smallest prime divisor of $p_1\dotsc p_n + 1$
negative_passages.docid: 112031
negative_passages.text: Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers.
negative_passages.docid: 67291
negative_passages.text: Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
negative_passages.docid: 24446
negative_passages.text: Prove that for each $n$, there exist $n$ consecutive integers, each of which is divisible by a perfect square larger than $1$.
negative_passages.docid: 97805
negative_passages.text: $n+1$ is a divisor of $\binom{2n} {n}$