query: The number $2^{3^n}+1$ is divisible by $3^{n+1}$ and is not divisible by $3^{n+2}$.
positive_passages.docid: 144132
positive_passages.text: The number $2^{3^n}+1$ is divisible by $3^{n+1}$ and not divisible by $3^{n+2}$.
negative_passages.docid: 46158
negative_passages.text: A positive integer is divisible by $3$ iff 3 divides the sum of its digits
negative_passages.docid: 96737
negative_passages.text: Geometric proof that a Pythagorean triple has a number divisible by $3$?
negative_passages.docid: 153281
negative_passages.text: Prove by induction that $3^{2n+3}+40n-27$ is divisible by 64 for all n in natural numbers
negative_passages.docid: 49096
negative_passages.text: How can I prove $5^{3^n}+1$ is divisible by $3^{n+1}$
negative_passages.docid: 43939
negative_passages.text: Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1.
negative_passages.docid: 122577
negative_passages.text: Prove that $(n^3 - 1)n^3(n^3 + 1)$ is divisible by $504$
negative_passages.docid: 101927
negative_passages.text: Induction: Prove that $4^{n+1}+5^{2 n - 1}$ is divisible by 21 for all $n \geq 1$.
negative_passages.docid: 48333
negative_passages.text: Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
negative_passages.docid: 66016
negative_passages.text: Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$
negative_passages.docid: 79219
negative_passages.text: Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
negative_passages.docid: 122019
negative_passages.text: Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$
negative_passages.docid: 2522
negative_passages.text: Why is $a^n - b^n$ divisible by $a-b$?
negative_passages.docid: 103784
negative_passages.text: Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer)
negative_passages.docid: 117167
negative_passages.text: $3^{n+1}$ divides $2^{3^n}+1$
negative_passages.docid: 117969
negative_passages.text: Suppose $n$ divides $3^n + 4^n$. Show that $7$ divides $n$.
negative_passages.docid: 1837
negative_passages.text: Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$